Center for the Advancement of Natural Discoveries using Light

Emission

Samvel Zakaryan

Electromagnetic waves in laminated structures and particle

bunching

Thesis

to take a candidate degree in physical and mathematical sciences 01.04.20

“Charged Particle Beam Physics and Accelerator Technology”

Scientific supervisor:

Doctor of Physical and Mathematical Sciences

Mikayel Ivanyan

Yerevan 2017

- 2 -

Contents

Introduction ............................................................................................. - 4 -

Chapter 1: Slow waves in laminated structures .................................. - 10 -

1.1 Introduction ..................................................................................................................... - 10 -

1.2 Slow waves in cylindrical structures ........................................................................ - 10 -

1.3 Attenuation parameter .................................................................................................. - 13 -

1.4 Longitudinal impedance and radiation pattern ...................................................... - 14 -

1.4. Slow waves in flat structures .................................................................................... - 16 -

1.5 Summary .......................................................................................................................... - 24 -

Chapter 2: Accelerating structures with flat geometry ....................... - 26 -

2.1 Introduction ..................................................................................................................... - 26 -

2.2 Multilayer flat structure ................................................................................................ - 27 -

2.3 Two-layer flat structure ................................................................................................ - 35 -

2.4 Impedance of two-layer symmetrical flat structure .............................................. - 37 -

2.5 Approximate analytical representation of impedance ......................................... - 47 -

2.6 Summary .......................................................................................................................... - 53 -

Chapter 3: Rectangular resonator with two-layer vertical walls ........ - 55 -

3.1 Introduction ..................................................................................................................... - 55 -

3.2 IHLCC cavity with ideally conductive walls ............................................................ - 56 -

3.3 Pure copper cavity. Measurements and interpretation ....................................... - 60 -

3.4 IHLCC cavity resonance frequency comparison with experimental results .. - 62 -

3.5 Summary .......................................................................................................................... - 67 -

Chapter 4: Electron bunch compression in single-mode structure ... - 68 -

4.1 Introduction ..................................................................................................................... - 68 -

4.2 Wakefields and Impedances ....................................................................................... - 69 -

4.3 Ballistic bunching .......................................................................................................... - 71 -

4.4 Energy modulation, bunch compression and microbunching in cold neutral

plasma ..................................................................................................................................... - 73 -

4.5 Energy modulation, bunch compression and microbunching in internally

coated metallic tube ............................................................................................................. - 81 -

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4.6 Summary .......................................................................................................................... - 88 -

Summary ................................................................................................ - 90 -

Acknowledgements ............................................................................... - 93 -

References ............................................................................................. - 94 -

- 4 -

Introduction

The investigation of new acceleration structures and high frequency, high brightness

coherent radiation sources is an important research area in modern accelerator physics [1-

7]. High brightness and high intensity radiation in 𝑇𝐻𝑧 region opens opportunities of

researches in wide range of areas [8-11]. European XFEL, SwissFEL accelerators are

constructed to obtain such radiation [12, 13]. To emit the 𝑇𝐻𝑧 radiation, one needs to

generate ultrashort, intensive bunches [12, 14-21]. When lengths of microbunches become

smaller than the synchronous mode wavelength (𝜎𝑧 < 𝜆), the large number of electrons

starts to radiate coherently and the intensity of the radiation field grows quadratically with

the number of particles [22].

The direct generation of sub-ps bunches in RF guns is limited due to technical

characteristics of photocathodes and lasers. Because of that reason, sub-ps bunches are

usually formed by initial long bunches shortening. One of the main principles to obtain

ultrashort bunches is to use SASE process [23, 24], which is widely used in modern

accelerators (FELs) [12, 13]. Also it is possible to use additional external laser field to have

a stronger microbunching []. The main principle to generate ultrashort bunches is to obtain

energy modulation within the bunch and convert it into charge density modulation [19]. For

the high energy bunches magnetic chicanes and for low energy bunches velocity bunching

are used to generate sub-ps bunches [25-29]. Energy modulation within the bunch can be

obtained via bunch interaction with its own radiated wakefield or wakefield emitted from

bunches in front of it []. For such a process disc loaded or dielectric loaded structures are

widely used [30-35]. Both structure types are characterized by the high order modes,

excited during the beam passage through them [35-46]. These high order modes play

parasitic role in particle acceleration processes.

Unlike disc and dielectric loaded structures, in the single mode structures like

plasma and circular waveguide with two-layer metallic walls (ICMT) there are no parasitic

high order modes [47-50]. In the dissertation the microbunching processes are studied in

single mode structure.

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Theoretically is shown that ICMT has a single slowly propagating high frequency

mode [40]. At appropriate structure geometry, a charged particle moving along the

structure axis radiates in the terahertz frequency region [50]. While the theoretical study of

ICMT shows that for certain conditions it is a single-mode structure, the experiment on it is

related to some technical difficulties. To serve as a high frequency single-mode structure,

ICMT requires high accuracy, 𝜇𝑚 – sub-𝜇𝑚 low conductivity internal coating, and an inner

diameter in the order of several 𝑚𝑚 [50], which is technically complex problem. Because

of these problems, it is preferable to do an experiment on a structure which is similar to

ICMT and is more simple in mechanical means. As such a structure, multilayer flat metallic

structure is considered. For appropriate parameters these two structures are similar, and it

is expected that two-layer flat structure should be single-mode, high frequency too.

The thesis consists of introduction, four chapters, summary and bibliography.

In introduction the short review of the actuality of the problems and the main results of the

thesis are given.

In the first chapter of the dissertation is devoted to the study of slow waves in

laminated structures. The study of high frequency accelerating structures is an important

issue for the development of future compact accelerator concepts [51, 52]. The laminated

structures are widely used in advanced accelerators to meet the technical specifications

like high vacuum performance, cure of static charge, reduction of the impedance, etc. [53].

The electrodynamic properties of two-layer structures, based on field matching technique,

have been studied in Refs. [54–57]. For the internally coated metallic tube (ICMT), with

coating thickness smaller than the skin depth, the longitudinal impedance has a narrow

band resonance at high frequency [57], which is conditioned by the synchronous 𝑇𝑀01

fundamental mode [50]. In the first chapter, the peculiarities of high frequency 𝑇𝑀01 mode,

slowly propagating in ICMT structure, is analyzed both numerically and analytically. The

TM modes dispersion curves and the 𝑇𝑀01 mode attenuation constant are given.

Dispersion relations of flat two-layer structure is investigated. The longitudinal impedance

and the radiation patterns for large aperture ICMT structures are discussed.

The main results of the first chapter are

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The single slow propagating mode properties in laminated structures are

studied. It is shown that two-layer cylindrical structure has single slowly traveling

mode (𝑇𝑀01) in 𝑇𝐻𝑧 frequency range.

Explicit expression for attenuation constant of two-layer structure’s 𝑇𝑀01 mode

is obtained.

Dispersion relations of flat two-layer structure are derived.

The second chapter of the thesis is devoted to study of electrodynamic properties

of multilayer two infinite metallic parallel plates. A bunch traveling through the structure

excites wakefields. The longitudinal component of the wakefield produces extra voltage for

the trailing particles in the bunch. The Fourier transformation of the wake potential for the

point driving charge is an impedance of the structure, which presents the excited

electromagnetic field in frequency domain [37]. For ultrarelativistic particles the impedance

is independent of the beam parameters and can describe a structure in frequency domain.

The field matching technique, based on matching the tangential components of the

electromagnetic fields at the borders of layers, is used to calculate electromagnetic fields.

Matrix formalism is developed to couple electromagnetic field tangential components in

two borders of layers. Fourier transformed electromagnetic fields excited by a relativistic

point charge are obtained analytically for two-layer unbounded two parallel infinite plates.

As a special cases of the above, electromagnetic fields of single-layer unbounded

structure and two-layer structure with perfectly conducting outer layer are obtained.

Longitudinal impedance of two-layer structure with outer perfectly conducting layer, excited

by an ultrarelativistic particle is calculated numerically. It is shown that for low conductivity

inner layer the driving particle radiation has a narrow-band resonance. The resonance

frequency is well described by the formula 𝑓𝑟𝑒𝑠 =𝑐

2𝜋√

1

𝑎 𝑑, where 𝑎 is a half-height of a

structure, 𝑑 is a layer thickness and 𝑐 is the velocity of light in vacuum. The longitudinal

wake potential is calculated and it is shown that it is a quasi-periodic function with the

period Δs given by the resonant frequency Δs =c

𝑓𝑟𝑒𝑠.

The main results of the second chapter are

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Method of calculating point charge excited electromagnetic fields in multilayer

infinite parallel plates is developed.

Expressions of radiation fields in two-layer parallel plates are derived analytically.

Longitudinal impedance and dispersion relations of symmetrical two-layer

parallel plates with perfectly conducting outer layer is derived.

Comparison of two-layer parallel plates’ electro-dynamic properties and ICMT’s

electro-dynamic properties are performed.

Conditions in which the point charge radiation in two-layer parallel plates has a

narrow-band resonance are obtained.

Wakefields generated by a point charge traveling through the center of the two-

layer parallel plates with outer perfectly conducting material are calculated.

In the third chapter theoretical study of rectangular cavity with laminated walls and

comparison with experimental results is presented. On the second chapter it is shown that

for appropriate parameters two-layered parallel infinite plates have a distinct, narrow band

resonance. In this chapter, a real, limited structure, which is similar to the parallel plates, is

studied. As such a structure a rectangular cavity with laminated horizontal walls is

considered. For the cavity with vertical dimensions much bigger than horizontals, the

electrodynamic characteristics are expected to be similar to the ones for parallel plates. In

the case of resonator, unlike the parallel plates’ case, one has a discrete set of

eigenfrequencies in all three degrees of freedom. To be able to distinguish the main

resonance from secondary ones, finite wall resonator model (perfectly conducting

rectangular cavity with inner low conductivity layers at the top and bottom walls) is

developed and resonances of the model are studied.

Comparison of resonant frequencies of resonator model with resonances of the test

copper cavity with inner germanium layers and parallel two-layer plates is done. Good

agreement between these results is obtained (Δ𝑓 𝑓𝑟𝑒𝑧 < 0.03⁄ ). Experimental results show

the presence of the fundamental resonance which is caused by the two-layer horizontal

cavity walls. The measured fundamental frequencies are close to those calculated for the

- 8 -

two-layer parallel plates, their values are subjected to the laws specific to this kind of

structures: they are decreasing with cavity height increase.

The main results of the third chapter are

The resonance properties of a copper rectangular resonator and a copper

resonator with an internal germanium coating are experimentally investigated

and theoretically substantiated.

Theoretical model of a rectangular cavity with horizontal laminated walls is

created and electro-dynamical properties of the model are studied.

The good correlation between resonance frequencies of the model and test

facility is fixed: all resonances of the test structure corresponds to the modes of

the model.

It is shown that for a resonator with vertical dimensions much bigger than

horizontal ones, the resonance frequencies are in good agreement with

resonances of two-layer infinite parallel plates.

The forth chapter of the dissertation is devoted to the generation of sub-ps bunches

via bunch compression and microbunching processes in single-mode structures.

Wakefields, generated by electron bunch, propagating through a structure, act back on the

bunch particles or the particles traveling behind it producing energy modulation or

transverse kick [36-38, 59-61]. For appropriate parameters, this process can lead to

energy modulation within the bunch, which than can be transformed into charge density

modulation. For non-ultrarelativistic (10 𝑀𝑒𝑉) electron bunches, the energy modulation

causes velocity modulation and for proper ballistic distance leads to bunch compression or

microbunching.

Single-mode structures, due to absence of high order parasitic modes, are ideal

candidates for this process. Plasma channel and internally coated metallic tube are

observed as an example of single-mode structures. Ballistic bunching method is used: the

bunch is considered rigid in a structure and only energy modulations occur, than in the drift

space the energy modulation leads to charge redistribution.

The main results of the forth chapter are

- 9 -

It is shown that the Gaussian distributed bunch interaction with single mode

structure leads to its compression. For appropriate bunch and structure

parameters 8 and 12 times bunch compression is reached for plasma and ICMT,

respectively.

It is shown that microbunching of non-Gaussian bunches is possible in single

mode structures. For parabolic distributed bunches 20 𝜇𝑚 and 3 𝜇𝑚

microbunches and for rectangular distributed bunches 40 𝜇𝑚 and 8 𝜇𝑚

microbunches can be formed for plasma and ICMT, respectively.

The relations between the bunch length, structures parameters and microbunch

formation distance are analyzed for various bunch shapes. The optimal

parameters for structures and drift space distances are obtained.

The main results of the dissertation are the following:

The single slowly propagating mode properties in laminated structures are

studied.

The electromagnetic fields of multilayer parallel plates are analytically derived.

Dispersion relations of two-layer parallel plates is derived and single resonance

frequency is obtained.

The narrow-band high frequency longitudinal impedance and longitudinal wake

function of two-layer parallel plates are calculated.

Resonance frequencies of rectangular cavity with horizontal double-layer

metallic walls are derived.

Good agreement between theoretical results and experiment is fixed.

It is shown that generation of ultrashort bunches (~10 𝜇𝑚 ) in single-mode

structures is possible via ballistic bunching.

The results of the study have been reported at international conference, during the

seminars at CANDLE Synchrotron Research Institute, Yerevan State University, DESY,

Paul Scherrer Institute and are published in the scientific journals.

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Chapter 1: Slow waves in laminated structures

1.1 Introduction

In this chapter the conditions in which slowly propagating free oscillations are

formed in cylindrical and flat two-layer metallic structures are studied. The importance of

establishing the presence of such oscillations is due to the fact that the radiation of a

moving particle in accelerating structures is synchronous (or is synphase) with its motion.

In particular, for an ultrarelativistic particle moving at the speed of light, the phase velocity

of the emitted wave is equal to the speed of light.

As is known, an arbitrary field, generated or propagating in a closed structure can

be expressed with the help of a superposition of the eigen-oscillations of a given structure.

The field generated by a particle must contain only synchronous components of its own

oscillations.

In this chapter, we consider mechanisms of the formation of a synchronous mode in

two-layer cylindrical and planar structures and demonstrate their connection with

longitudinal impedances. In particular, for a plane structure, a relationship between the

eigenvalues of the synchronous mode and the poles of the impedance integral is

established.

1.2 Slow waves in cylindrical structures

Consider a round metallic two-layer hollow pipe of inner radius 𝑎 with perfectly

conducting outer layer and the metallic inner layer of conductivity 𝜎 and thickness 𝑑 = 𝑏 −

𝑎 (Figure 1.1).

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Figure 1.1: The geometry of the internally coated metallic tube.

In axially symmetric geometry, the frequency domain non zero electromagnetic field

components (𝐸𝑧, 𝐸𝑟, 𝐻𝜃) of the axially symmetric monopole TM modes are proportional to

𝑒𝑥𝑝(𝑗𝛽0𝑛𝑧 − 𝑗𝜔𝑡) , where 𝛽0𝑛 = √𝑘2 − 𝜈0𝑛

2 and 𝜈𝑛 are the longitudinal and radial

propagation constants, respectively, 𝑘 = 𝜔 𝑐⁄ is the wave number, 𝜔 is the frequency and 𝑐

is the velocity of light. The dispersion relation for the modes is derived by the matching of

electromagnetic field tangential components (𝐸𝑧, 𝐻𝜃) at the layers boundaries. Omitting the

factor 𝑒𝑥𝑝(𝑗𝛽0𝑛𝑧 − 𝑗𝜔𝑡), the tangential components of electromagnetic fields in vacuum

(𝑟 < 𝑎) and metallic wall (𝑎 ≤ 𝑟 ≤ 𝑏) regions are given as

𝐸𝑧(𝑟, 𝑧) = 𝐴𝐽0(𝜈0𝑛𝑟)

𝐻𝜃(𝑟, 𝑧) = 𝐴𝑗𝜔𝜀0

𝜈0𝑛𝐽1(𝜈0𝑛𝑟)

𝐸𝑧(𝑟, 𝑧) = 𝐵𝐽0(𝜆0𝑛𝑟) + 𝐶𝐻0(1)(𝜆0𝑛𝑟)

𝐻𝜃(𝑟, 𝑧) =𝑗𝜔𝜀1

𝜆0𝑛[𝐵𝐽1(𝜆0𝑛𝑟) + 𝐶𝐻1

(1)(𝜆0𝑛𝑟)]

for

𝑟 < 𝑎

𝑎 ≤ 𝑟 ≤ 𝑏

(1.1)

where 𝜆0𝑛 = √𝜈0𝑛2 − 𝜒2, 𝜒 = 𝑗 √2 𝛿

⁄ is the radial propagation constant in metallic internal

layer, 𝛿 = 𝑐 (2휀0

𝜎𝜔⁄ ) is the skin depth, 휀0 and 휀1 = 휀0 +𝑗𝜎𝜔⁄ are the vacuum and metal

dielectric constants, respectively, 𝐽0,1(𝑥), 𝐻0,1(1)(𝑥) are the zero and first order Bessel and

Hankel functions respectively.

The unknown coefficients 𝐴, 𝐵, 𝐶 are defined by the matching of tangential

components (𝐸𝑧 , 𝐻𝜃 ) at (𝑟 = 𝑎) and the vanishing of the electric field 𝐸𝑧 at the perfect

- 12 -

conducting outer wall (𝑟 = 𝑏). Non-zero solution is provided by zero determinant of three

linear algebraic equations system, which in its turn defines the eigenvalue equation.

𝜀0𝐽1(𝜈0𝑛𝑎)

𝜀1𝐽0(𝜈0𝑛𝑎)=𝜈0𝑛𝐽1(𝜆0𝑛𝑎)𝐻0

(1)(𝜆0𝑛𝑏)−𝐽0(𝜆0𝑛𝑏)𝐻1

(1)(𝜆0𝑛𝑎)

𝜆0𝑛𝐽0(𝜆0𝑛𝑎)𝐻0(1)(𝜆0𝑛𝑏)−𝐽0(𝜆0𝑛𝑏)𝐻0

(1)(𝜆0𝑛𝑎)

(1.2)

In high frequency range, where the inequalities |𝜒| ≫ |𝜈0𝑛| (|𝜆0𝑛| ≈√2

𝛿⁄ ), |𝜆0𝑛|𝑎 ≫

1 and 𝛿 ≪ 𝑎 are hold, Eq. 1.2 is modified to

𝜀0𝐽1(𝜈0𝑛𝑎)

𝜈0𝑛𝐽0(𝜈0𝑛𝑎)=

𝜀1

𝜆0𝑛cot(𝜆0𝑛𝑑) (1.3)

For the inner metallic layer thickness 𝑑, less than the skin depth 𝛿 (𝑑 ≪ 𝛿), and not

very high frequencies 𝜔 ≪ 𝜎휀0⁄ (휀1 ≈

𝑗𝜎𝜔⁄ ), the Eq. 1.3 is expressed as

1

𝜈0𝑛𝑎

𝐽1(𝜈0𝑛𝑎)

𝐽0(𝜈0𝑛𝑎)=

1

𝑘2𝑎𝑑 (1.4)

In the frequency range of interest (THz), the inequality 𝜔 ≪ 𝜎휀0⁄ holds well for

practically all metals (for copper, e.g. 𝜎 휀0⁄ ≈ 105 𝑇𝐻). The frequency range of applicability

for expression (1.4) is given as 𝑑 ≪ 𝛿 ≪ 𝑎.

The analysis of the Eq. 1.4 shows that the radial propagation constants 𝜈0𝑛 of high

𝑇𝑀0𝑛 (𝑛 > 1) modes are purely real and nonzero. Therefore, the high order modes phase

velocity 𝜈𝑝ℎ =𝑐𝑘𝛽0𝑛⁄ is higher than velocity of light. For fundamental 𝑇𝑀01 mode, the

parameter 𝜈01 vanishes (𝜈01 = 0) at frequency 𝑘0 = √2/𝑎𝑑 (𝜈𝑝ℎ = 𝑐), 𝜈01 is purely real

for frequencies 𝑘 < 𝑘0 (𝜈𝑝ℎ > 𝑐) and 𝜈01 is purely imaginary for frequencies 𝑘 > 𝑘0 (𝜈𝑝ℎ <

𝑐). Thus, the structure under consideration is characterized by a single slowly propagating

𝑇𝑀01mode at high frequency. Figure 1.2 shows the dispersion curves for the fundamental

𝑇𝑀01 and high order 𝑇𝑀0𝑛 (𝑛 > 1) modes. As is seen, only the fundamental mode crosses

the synchronous line 𝛽 = 𝑘 (𝜈𝑝ℎ = 𝑐) at a resonant frequency of 𝑘 = 𝑘0 = √2/𝑎𝑑.

- 13 -

Figure 1.2: Dispersion curves for fundamental 𝑇𝑀01 and high order 𝑇𝑀0𝑛 (𝑛 > 1) modes.

1.3 Attenuation parameter

To evaluate the 𝑇𝑀01 mode attenuation constant 𝛼01 = 𝐼𝑚(𝛽01) for thin inner layer

(𝑑 ≪ 𝛿), the second term in expansion of formula 1.3 should be taken into account, i.e.

cot(𝜒) ≈ 1 𝜒⁄ −𝜒3⁄ ,

𝐽1(𝜈01𝑎)

𝜈01𝑎𝐽0(𝜈01𝑎)=

𝑘02

2𝑘2− 𝑗

𝜂

𝑘𝑎 (1.5)

where 휂 = (𝜍 + 𝜍−1)/√3 , 𝜍 = 𝑍0𝑑𝜎/√3 . The numerical solutions for real and

imaginary parts of 𝑇𝑀01 mode transverse propagation constant are presented on Figure

1.3 (𝑑 𝑎⁄ = 10−3 , 𝜍 = 1 ). For small arguments of Bessel functions |𝜈01|𝑎 ≪ 1 one can

obtain the following approximate expression

𝜈012 ≈

16

𝑎2(−

1

2+

𝑘02

2𝑘2− 𝑗

𝜂

𝑘𝑎) (1.6)

Note, that the real and imaginary parts of transverse propagation constant 𝜈01 at

the resonance frequency 𝑘 = 𝑘0 = √2𝑎𝑑⁄ are given as

𝑅𝑒(𝜈01) = −𝐼𝑚(𝜈01) =1

𝑎√8𝜂

𝑘0𝑎 (1.7)

- 14 -

and the relation |𝜈01|𝑎 ≪ 1 well hold around the resonance frequency. For high frequency

𝑘 ≫ |𝜈01|, the longitudinal propagation constant 𝛽01 = √𝑘2 − 𝜈012 is given as

𝛽 ≈ 𝑘 +4

𝑘3𝑎2(𝑘2 − 𝑘0

2) +8𝜂

𝑎(𝑘𝑎)2 (1.8)

The attenuation constant at resonance frequency (𝑘 = 𝑘0) is then given as 𝛼01 =

4𝑑휂/𝑎2.

Figure 1.3: Real (solid) and imaginary (dashed) parts of 𝑇𝑀01 mode transverse eigenvalue 𝜈 =

𝜈01𝑎.

1.4 Longitudinal impedance and radiation pattern

Consider a relativistic charge traveling along the axis of a round metallic two-layer

hollow pipe (Fig. 1.1). The excited nonzero electromagnetic field components (𝐸𝑧, 𝐸𝑟, 𝐻𝜃)

of the axially symmetric monopole mode in the vacuum region can be derived based on

the field matching technique [53]. A good analytical approximation for metallic type two-

layer tube longitudinal impedance can be obtained in high frequency range, when the skin

depths 𝛿 are much smaller than the tube radius a, i.e. 𝛿 ≪ 𝑎 [57].

𝑍∥0 = 𝑗

𝑍0

𝜋𝑘𝑎2(1 +

2

𝑎

𝜀

𝜀0𝜒𝑐𝑡ℎ(𝜒𝑑))

−1

(1.9)

- 15 -

where 𝑍0 = 120𝜋 Ω is the impedance of free space. For the layer thickness 𝑑 smaller with

respect to skin depth 𝛿 (𝑑 ≪ 𝛿) and not very high frequencies (𝜔 ≪ 𝜎/휀0) the impedance

can be presented in a form of the impedance of parallel resonance circuit

𝑍∥0 = 𝑅 [1 + 𝑗𝑄 (

𝜔0

𝜔−

𝜔

𝜔0)]−1

(1.10)

where 𝜔0 = 𝑐√2/𝑎𝑑 is the resonant frequency, 𝑅 = 𝑍0(2𝜋𝑎휂)−1 is the shunt impedance,

𝑄 = 𝜔0𝑎(2𝑐휂)−1 is the quality factor.

Figure 1.4: Longitudinal impedance for Cu, LCM (solid) and Cu-LCM (dashed) tubes.

Figure 1.4 presents the field matching based exact numerical simulations of

longitudinal impedance for copper (Cu, 𝜎1 = 5.8 × 107 Ω−1𝑚−1) tube internally coated by

the low conductivity metal (LCM, 𝜎1 = 5 × 103 Ω−1𝑚−1 ) of 𝑑 = 1 𝜇𝑚 and 𝑑 = 0.25 𝜇𝑚

thickness. The radius of tube is 𝑎 = 1 𝑐𝑚. As is seen, in the transition region for thin LCM

layer the impedance modified to narrow band resonance at the frequency 𝑓0 = 𝜔0/2𝜋,

corresponding to 𝑓0 = 0.675 𝑇𝐻𝑧 for 𝑑 = 1 𝜇𝑚 and 𝑓0 = 1.35 𝑇𝐻𝑧 for 𝑑 = 0.25 𝜇𝑚 . The

impedance has a different nature for single and two-layer tubes. It has a broadband

maximum for the single layer tube and narrow band resonance at high frequency ω for

two-layer tube. An important feature is that the resonance is observed when the inner layer

thickness is less than the layer skin depth. Thus, the resonance is conditioned by the

interference of scattered electromagnetic fields from the inner and outer layers.

- 16 -

The radiation pattern at the open end of the waveguide can be evaluated by using

from near to far fields transformation technique.

Figure 1.5: Radiation pattern for frequencies 𝑓 = 1.35 𝑇𝐻𝑧 (solid) and 𝑓 = 4.7 𝑇𝐻𝑧 (dashed).

Fig. 1.5 presents the angular distribution of the normalized radiation intensity with

respect to waveguide axis 휃 = 0 for the internal layer thickness of 𝑑 = 0.25 𝜇𝑚 and

resonant frequencies 𝑓 = 4.7 𝑇𝐻𝑧 (𝑎 = 1 𝑚𝑚) and 𝑓 = 1.35 𝑇𝐻𝑧 (𝑎 = 1 𝑐𝑚). The radiation

pattern is zero at 휃 = 0 and reaches its maximum value at 휃 = 𝑎𝑟𝑐𝑠𝑖𝑛(2.3√𝑑/2𝑎).

1.4. Slow waves in flat structures

In this section, we determine the dispersion relations that determine the

eigenfrequencies of the electromagnetic oscillations of the structure consisting of two

infinite parallel planes, each of which is covered from the inside by a thin conducting layer.

The metal layers are identical: with a thickness 𝑑 and a conductivity 𝜎. The permittivity of

the layers is characterized by the expression 휀 = 휀′ + 𝑗 𝜎 𝜔⁄ , where 휀′ the static permittivity

of the metal (for most metals 휀′ = 휀0, where 휀0 is the permittivity of vacuum, for germanium

휀′ = 16휀0) and 𝜔 is a frequency. The distance between the inner surfaces of the plates 2𝑎.

The solution for the natural electromagnetic oscillations (eigen-oscillations) of the

structure is sought by the method of partial regions. In the case under consideration, we

can distinguish three regions: the region of the upper (𝑎 ≤ 𝑦 ≤ 𝑎 + 𝑑, 𝑖 = 1) and lower

(−𝑎 − 𝑑 ≤ 𝑦 ≤ 𝑎, 𝑖 = −1) metal layer and the vacuum region between the plates (−𝑎 ≤

- 17 -

𝑦 ≤ 𝑎, 𝑖 = 0). In the outer ideally conducting layers, the fields and currents are absent.

Since all three regions have finite dimensions along the transverse direction, in each of

them can propagate quasi-plane waves with both damped and increasing amplitudes.

Thus, in each of the three regions, the fields can be written in an identical form. In

particular, electrical components in each of the three regions (𝑖 = 0, ±1) are written as:

𝐸𝑥,𝑦,𝑧(𝑖)

= 𝐸1,𝑥,𝑦,𝑧(𝑖)

+ 𝐸2,𝑥,𝑦,𝑧(𝑖)

(1.11)

with

𝐸1,2,𝑥,𝑦,𝑧(𝑖) (𝑟) = ∫ ∫ �̃�1,2,𝑥,𝑦,𝑧

(𝑖)𝑑𝑘𝑥𝑑𝑘𝑧𝑑𝑘

∞

−∞

∞

−∞ (1.12)

where

�̃�1,2,𝑥(𝑖)

= 𝐴1,2,𝑥 {𝑆ℎ (𝑘𝑦

(𝑖)𝑦)

𝐶ℎ (𝑘𝑦(𝑖)𝑦)} 𝑒𝑗(𝑘𝑥𝑥+𝑘𝑧𝑧−𝜔𝑡)

�̃�1,2,𝑦(𝑖)

= 𝐴1,2,𝑦 {𝐶ℎ (𝑘𝑦

(𝑖)𝑦)

𝑆ℎ (𝑘𝑦(𝑖)𝑦)} 𝑒𝑗(𝑘𝑥𝑥+𝑘𝑧𝑧−𝜔𝑡)

�̃�1,2,𝑧(𝑖)

= 𝐴1,2,𝑧 {𝑆ℎ (𝑘𝑦

(𝑖)𝑦)

𝐶ℎ (𝑘𝑦(𝑖)𝑦)} 𝑒𝑗(𝑘𝑥𝑥+𝑘𝑧𝑧−𝜔𝑡)

(1.13)

Here 𝐴1,2,𝑥,𝑦,𝑧 are arbitrary weight coefficients (amplitudes) and 𝜔 = 𝑘𝑐 . The

magnetic field components are determined from electrical field components with the help

of Maxwell equations:

�⃗⃗�1,2(𝑖)(𝑟) =

1

𝑗𝜔𝑟𝑜𝑡 �⃗⃗�±

(𝑖)(𝑟) (1.14)

whence

- 18 -

�̃�1,2,𝑥(𝑖) =

1

𝑗𝜔(𝑘𝑦

(𝑖)𝐴1,2,𝑧(𝑖) − 𝑗𝑘𝑧𝐴1,2,𝑦

(𝑖) ) {𝐶ℎ(𝑘𝑦

(𝑖)𝑦)

𝑆ℎ(𝑘𝑦(𝑖)𝑦)} 𝑒𝑗(𝑘𝑥𝑥+𝑘𝑧𝑧−𝜔𝑡)

�̃�1,2,𝑦(𝑖) = −

1

𝜔(𝑘𝑥𝐴1,2,𝑧

(𝑖) − 𝑘𝑧𝐴1,2,𝑥(𝑖) ) {

𝑆ℎ(𝑘𝑦(𝑖)𝑦)

𝐶ℎ(𝑘𝑦(𝑖)𝑦)

} 𝑒𝑗(𝑘𝑥𝑥+𝑘𝑧𝑧−𝜔𝑡)

�̃�1,2,𝑥(𝑖)

= −1

𝑗𝜔(𝑘𝑦

(𝑖)𝐴1,2,𝑥(𝑖)

− 𝑗𝑘𝑧𝐴1,2,𝑦(𝑖) ) {

𝐶ℎ(𝑘𝑦(𝑖)𝑦)

𝑆ℎ(𝑘𝑦(𝑖)𝑦)

} 𝑒𝑗(𝑘𝑥𝑥+𝑘𝑧𝑧−𝜔𝑡)

(1.15)

The relationship between the amplitudes 𝐴1,2,𝑧(𝑖)

and 𝐴1,2,𝑥(𝑖) , 𝐴1,2,𝑦

(𝑖) is determined from

the Maxwell equation

𝑑𝑖𝑣�⃗⃗�1,2(𝑖)= 0 (1.16)

whence

𝐴1,2,𝑧(𝑖)

=1

𝑘𝑧(𝑗𝑘𝑦

(𝑖)𝐴1,2,𝑥(𝑖)

− 𝑘𝑥𝐴1,2,𝑥(𝑖)

) (1.17)

To determine the relationship between wave numbers and frequency we use the

identity, which follows from Eq. 1.13

𝑟𝑜𝑡 𝑟𝑜𝑡�⃗⃗�1,2(𝑖)= (𝑘𝑥

2 − (𝑘𝑦(𝑖))2+ 𝑘𝑧

2) �⃗⃗�1,2(𝑖)(𝑘𝑥, 𝑘𝑧, 𝑘, 𝑟) (1.18)

and the equation, which is a consequence of Maxwell's equations:

𝑟𝑜𝑡 𝑟𝑜𝑡�⃗⃗�±(𝑖)= 𝑘2휀𝑖

′𝜇𝑖′�⃗⃗�±(𝑖)

(1.19)

with 휀𝑖′, 𝜇𝑖

′ relative electric and magnetic permeability in the respective regions (휀±1′ = 1 +

𝑗 𝜎 휀0𝜔, 𝜇±1′ = 1⁄ ; 휀0

′ = 𝜇0′ = 1), and 𝑘 = 𝜔 𝑐⁄ .

Comparing Eq. 1.18 and Eq. 1.19, we obtain

𝑘𝑦(𝑖)= √−𝑘2휀𝑖

′𝜇𝑖′ + 𝑘𝑥

2 + 𝑘𝑧2 (1.20)

For inner (vacuum) part of structure

𝑘𝑦(0)= √−𝑘2 + 𝑘𝑥

2 + 𝑘𝑧2 (1.21)

- 19 -

To construct the dispersion relations (relationship between 𝑘𝑥 and k) one should set

up a system of linear equations based on fields matching at the boundaries of the structure

areas:

1. 𝐸𝑥(1)= 0, 𝐸𝑧

(1)= 0 at 𝑦 = 𝑎 + 𝑑

(1.22)

2. 𝐸𝑥(1) = 𝐸𝑥

(0), 𝐸𝑧(1) = 𝐸𝑧

(0); 𝐵𝑥(1) = 𝐵𝑥

(0), 𝐵𝑧(1) = 𝐵𝑧

(0) at 𝑦 = 𝑎

3. 𝐸𝑥(−1) = 𝐸𝑥

(0), 𝐸𝑧(−1) = 𝐸𝑧

(0); 𝐵𝑥(−1) = 𝐵𝑥

(0), 𝐵𝑧(−1) = 𝐵𝑧

(0) at 𝑦 = −𝑎

4. 𝐸𝑥(−1)

= 0, 𝐸𝑧(−1)

= 0 at 𝑦 = −𝑎 − 𝑑

As a result, a system of 12 equations containing 12 unknown amplitudes is obtained.

Dispersion relations are determined by equating to zero the determinant of this system,

which takes the form consisting of the product of four factors:

𝐷 = ∏ 𝑇𝑖4𝑖=1 = 0 (1.23)

where

𝑇1 = 𝑘𝑦(1)𝐶ℎ (𝑑𝑘𝑦

(1))𝑆ℎ (𝑎𝑘𝑦(0)) + 𝑘𝑦

(0)𝑆ℎ (𝑑𝑘𝑦(1))𝐶ℎ (𝑎𝑘𝑦

(0))

𝑇2 = 𝑘𝑦(1)𝐶ℎ (𝑑𝑘𝑦

(1))𝐶ℎ (𝑎𝑘𝑦(0)) + 𝑘𝑦

(0)𝑆ℎ (𝑑𝑘𝑦(1))𝑆ℎ (𝑎𝑘𝑦

(0))

𝑇3 = 𝑘𝑦(0)(𝑘𝑦(1)2

− 𝑘𝑥2 − 𝑘𝑧

2)𝐶ℎ (𝑑𝑘𝑦(1)) 𝑆ℎ (𝑎𝑘𝑦

(0)) + 𝑘𝑦

(1)(𝑘𝑦(0)2

− 𝑘𝑥2 − 𝑘𝑧

2) 𝑆ℎ (𝑑𝑘𝑦(1))𝐶ℎ (𝑎𝑘𝑦

(0))

𝑇4 = 𝑘𝑦(0) (𝑘𝑦

(1)2 − 𝑘𝑥2 − 𝑘𝑧

2)𝐶ℎ (𝑑𝑘𝑦(1))𝐶ℎ (𝑎𝑘𝑦

(0)) + 𝑘𝑦(1) (𝑘𝑦

(0)2 − 𝑘𝑥2 − 𝑘𝑧

2)𝑆ℎ (𝑑𝑘𝑦(1)) 𝑆ℎ (𝑎𝑘𝑦

(0))

(1.24)

Taking into account 𝑘𝑦(𝑖) = √−𝑘2휀𝑖

′𝜇𝑖′ + 𝑘𝑥2 + 𝑘𝑧2 from Eq. 1.18, instead of 𝑇3 and 𝑇4

we can write

𝑇3 = −𝑘𝑦(0)𝑘2휀𝑖

′𝜇𝑖′𝐶ℎ(𝑑𝑘𝑦

(1))𝑆ℎ(𝑎𝑘𝑦(0)) − 𝑘𝑦

(1)𝑘2𝑆ℎ(𝑑𝑘𝑦

(1))𝐶ℎ(𝑎𝑘𝑦(0))

𝑇4 = −𝑘𝑦(0)𝑘2휀𝑖

′𝜇𝑖′𝐶ℎ(𝑑𝑘𝑦

(1))𝐶ℎ(𝑎𝑘𝑦(0)) − 𝑘𝑦

(1)𝑘2𝑆ℎ(𝑑𝑘𝑦(1))𝑆ℎ(𝑎𝑘𝑦

(0)) (1.25)

The equality to zero of each of the factors (𝑇𝑖 = 0, 𝑖 − 1,2,3,4) determines an

independent system of natural oscillations of the structure.

The radiation generated by a particle is synchronous with its motion. The spectral

components of its radiation field have a phase velocity equal to the speed of light 𝑣𝑝ℎ =

𝜔 𝑘⁄ = 𝑐. In this case, the direction of motion of the particle (direction along the z axis) is

- 20 -

distinguished. In the case of natural electromagnetic oscillations, determined in the

absence of a particle or some other source of radiation, the directions along both

horizontal axes �⃗� and 𝑧 are equivalent. This circumstance manifests itself in a symmetrical

form of the dependence of the frequency 𝑘 = 𝜔 𝑐⁄ on the wave numbers 𝑘𝑥 and 𝑘𝑧 (1.21):

𝑘 = √𝑘𝑥2 + 𝑘𝑧

2 − 𝑘𝑦(0)2

(1.26)

The phase velocity of the eigenfunctions of the structure has the form:

𝑣𝑝ℎ = 𝜔 𝑘𝑧⁄ =𝑘

𝑘𝑧𝑐 =

√𝑘𝑥2+𝑘𝑧

2−𝑘𝑦(0)2

𝑘𝑧𝑐 (1.27)

As can be seen from Eq. 1.27, the phase velocity of the natural oscillations is equal

to the velocity of the particle (the speed of light in a vacuum) under condition 𝑘 = 𝑘𝑧. The

amplitudes 𝐴1,2,𝑥,𝑦,𝑧 are then written in a form containing the Dirac delta function: 𝐴1,2,𝑥,𝑦,𝑧 =

�̃�1,2,𝑥,𝑦,𝑧𝛿(𝑘 − 𝑘𝑧). Taking into account 𝑘 = 𝑘𝑧 the vertical wave numbers 𝑘𝑦(𝑖)

will take the

form 𝑘𝑦(𝑖) = √𝑘2(1 − 휀𝑖

′𝜇𝑖′) + 𝑘𝑥2 + 𝑘𝑧2 for 𝑖 = ±1 (in layers) and 𝑘𝑦

(0) = 𝑘𝑥 at 𝑖 = 0 (in the

inner vacuum region)

Thus, for synchronous eigenmodes generated by a particle, instead of (1.24, 1.25),

one can write (by replacing 𝑘𝑦(0)

with 𝑘𝑥)

𝑇1 = 𝑘𝑦(1)𝐶ℎ (𝑑𝑘𝑦

(1)) 𝑆ℎ(𝑎𝑘𝑥) + 𝑘𝑥𝑆ℎ (𝑑𝑘𝑦

(1)) 𝐶ℎ(𝑎𝑘𝑥)

𝑇2 = 𝑘𝑦(1)𝐶ℎ (𝑑𝑘𝑦

(1))𝐶ℎ(𝑎𝑘𝑥) + 𝑘𝑥𝑆ℎ (𝑑𝑘𝑦

(1)) 𝑆ℎ(𝑎𝑘𝑥)

𝑇3 = −𝑘𝑥𝑘2휀𝑖′𝜇𝑖′𝐶ℎ (𝑑𝑘𝑦

(1)) 𝑆ℎ(𝑎𝑘𝑥) − 𝑘𝑦

(1)𝑘2𝑆ℎ (𝑑𝑘𝑦

(1)) 𝐶ℎ(𝑎𝑘𝑥)

𝑇4 = −𝑘𝑥𝑘2휀𝑖′𝜇𝑖′𝐶ℎ (𝑑𝑘𝑦

(1))𝐶ℎ(𝑎𝑘𝑥) − 𝑘𝑦

(1)𝑘2𝑆ℎ (𝑑𝑘𝑦

(1)) 𝑆ℎ(𝑎𝑘𝑥)

(1.28)

Let us compare the obtained expressions 1.28 with the denominators of the

impedance (Eq. 2.45) obtained in Chapter 2 writing them in a form convenient for

comparison

- 21 -

𝑅1𝐶

𝑅2𝐶} = ±{

𝑘𝑦(1)

𝑘𝑥𝑘2휀𝑖′𝜇𝑖′}𝐶ℎ(𝑘𝑦

(1)𝑑)𝐶ℎ(𝑘𝑥𝑎1) ± {𝑘𝑥

𝑘𝑦(1)𝑘2} 𝑆ℎ(𝑘𝑦

(1)𝑑)𝑆ℎ(𝑘𝑥𝑎1)

𝑅1𝑆

𝑅2𝑆} = ±{

𝑘𝑦(1)

𝑘𝑥𝑘2휀𝑖′𝜇𝑖′}𝐶ℎ(𝑘𝑦

(1)𝑑)𝑆ℎ(𝑘𝑥𝑎1) ± {𝑘𝑥

𝑘𝑦(1)𝑘2} 𝑆ℎ(𝑘𝑦

(1)𝑑)𝐶ℎ(𝑘𝑥𝑎1)

(1.29)

The comparison gives a complete coincidence of the expressions 1.28 and 1.29

𝑇1 = 𝑅1𝑆, 𝑇2 = 𝑅1

𝐶 , 𝑇3 = 𝑅2𝑆, 𝑇4 = 𝑅2

𝐶 (1.30)

Thus, the complex roots of the dispersion equations (𝑇𝑖 = 0, 𝑖 = 1,2,3,4) are the

poles of impedance function on the complex plane 𝑘𝑥. When the particle moves along the

symmetry plane of the structure (Δ𝑦 = 0), only two dispersion equations are realized: 𝑇2 =

0 and 𝑇4 = 0. With a parallel offset of the particle from the horizontal plane of symmetry

(Δ𝑦 ≠ 0), all four dispersion equations are realized: 𝑇𝑖 = 0, 𝑖 = 1,2,3,4.

In the approximation 𝑘𝑥2 ≪ 𝑍0𝜎𝑘𝑧 or 𝑘𝑦

(1) ≈ √−𝑗𝑍0𝜎𝑘𝑧 , (in the cylindrical case this

approximation is equivalent to approximation for the permittivity of the metal 휀 ≈ 𝑗 𝜎 𝜔⁄ ,

which is valid for not very large frequencies), the expressions (1.28) or (1.29) become

analytic on the complex plane 𝑘𝑥 , which allows us to calculate directly the integral

functions through which the impedance is expressed using the residue theorem [62].

The final form of the second and fourth equations (1.28) is

𝑘𝑥𝑇ℎ(𝑎𝑘𝑥) = −√−𝑗𝑍0𝜎𝑘𝐶𝑡ℎ(𝑑√−𝑗𝑍0𝜎𝑘)

𝑇ℎ(𝑎𝑘𝑥)

𝑘𝑥=√−𝑗𝜎𝑍0𝑘

𝑘2𝐶𝑡ℎ(𝑑√−𝑗𝜎𝑍0𝑘)

(1.31)

At a fixed frequency, the right-hand sides of Eq. 1.31 are constant (independent of

the wave number 𝑘𝑥 to be determined). Equations (1.31) can be expressed through

dimensionless parameters �̃�𝑥 = 𝑎𝑘𝑥 , �̃� = 𝑘𝑑, �̃� = 𝑑 𝑎⁄ and 𝜉 = 𝑍0𝜎𝑑:

𝐶𝑡ℎ(�̃�𝑥)

�̃�𝑥= −�̃�

𝑡ℎ(√−𝑗𝜉�̃�)

√−𝑗𝜉�̃� (1.32a)

𝑇ℎ(�̃�𝑥)

�̃�𝑥= �̃�√−𝑗𝜉�̃� 𝐶𝑡ℎ (√−𝑗𝜉�̃�) �̃�2⁄ (1.32b)

- 22 -

Fig.1.6. Eigenvalues of a planar two-layer structure. Solutions of equation 1.32a (left) and 1.32b

(right) for the different values of parameter 𝜉 (𝜉 = 1, 2 𝑎𝑛𝑑 5) at a resonance frequency �̃� =

𝑘𝑟𝑒𝑧�̃� = √𝑑 𝑎⁄ , �̃� = 10−4.

Fig.1.7. Eigenvalues of a planar two-layer structure. Solutions of equation 1.32a (left) and 1.32b

(right) in the case of 𝜉 = 1 for the several values of frequency: �̃� = 𝑘𝑟𝑒𝑧�̃� = √𝑑 𝑎⁄ , �̃� = 0.5√𝑑 𝑎⁄ <

𝑘𝑟𝑒𝑧�̃� and �̃� = 1.5√𝑑 𝑎⁄ > 𝑘𝑟𝑒𝑧�̃�; �̃� = 10−4.

Figures 1.6 and 1.7 show the eigenvalue distributions of the synchronous mode of a

two-layered planar structure. On the horizontal and vertical axes, respectively, the

imaginary and real components of the eigenvalues are plotted on the graphs. The graphs

are the result of the exact solution of Equations (1.32a, b). For small right-hand sides of

equations (1.32), their asymptotic solution can be represented as the sum of the solution

for ideally conducting plates (𝑘𝑥0𝑎 = 𝑗 𝜋 2⁄ + 𝑗𝜋(𝑛 − 1) for Equation (1.32a) and 𝑘𝑥0𝑏 = 𝑗𝜋𝑛

for Equation (1.32b), 𝑛 = 1,2,4, … ) and a small additive 𝛼𝑎(𝑏) , taking into account the

presence of the inner coating:

- 23 -

𝑘𝑥𝑎(𝑏) = 𝑗𝑘𝑥0𝑎(𝑏) + 𝛼𝑎(𝑏), 𝛼𝑎(𝑏) = −𝑗𝑑𝑎(𝑏)𝑘0𝑎(𝑏)

1±𝑑𝑎(𝑏) (1.33)

where

𝑑𝑎 = �̃� 𝑡ℎ (√−𝑗𝜉�̃�) √−𝑗𝜉�̃�⁄ , 𝑑𝑏 = �̃�√−𝑗𝜉�̃� 𝐶𝑡ℎ (√−𝑗𝜉�̃�) �̃�2⁄ (1.34)

The smallness of the parameters 𝑑𝑎(𝑏) (1.34) is conditioned by not only due to the

smallness of the relative thickness of the layer �̃�, but also to the smallness of the other

factors included in this parameter. Thus, for �̃� = 10−4 and 𝜉 = 1 at the resonant frequency

�̃� = √𝑑 𝑎⁄ , the parameter 𝑑𝑏 is of the order of unity (⌊𝑑𝑏⌋ ≈ 1).

The eigenvalues 𝑘𝑥𝑎 (solutions of the equation (1.32a)) within the parameters

shown in Figures 1.6, 1.7 (left), are approximated rather well by the formula (1.33). They

are characterized by imaginary components closed enough to the eigenvalues of the

structure consisting of two parallel ideally conducting planes (𝐼𝑚𝑘𝑥𝑎 ≈ 𝑘𝑥𝑎0), and by small

real additions, whose values decrease with increasing parameter 𝜉 (Fig.1.6, left). We note

that for fixed values of the parameter 𝜉 , small real corrections increase linearly with

increasing imaginary components (Fig. 1.6, 1,7, left) on x axis and decreases with

increasing frequency (Fig. 1,7, left).

The behavior of sequences of eigenvalues 𝑘𝑏 (solutions of equation (1.32б)) is

much more complicated. Their real components are two or three orders of magnitude

smaller than the corresponding roots of equation (1.32a). For small 𝜉 (𝜉 = 1 and 𝜉 = 2 on

Figure 1.6, right) they decrease exponentially with increasing imaginary components,

deposited on the horizontal axis. With an increase of the parameter 𝜉, the lower values

begin to increase to a certain limit, and then the increase goes down (𝜉 = 5 on Figure 1.6,

right). In contrast to the eigenvalues of 𝑘𝑎 (solutions of equation (1.32a)), there is a

general increase of the real components of eigenvalues with increasing of parameter 𝜉

(Figure 1.6, right). The reverse order is also valid for fixed values of 𝜉: lower frequencies

correspond to smaller values of the real components of the eigenvalues 𝑘𝑏. In contrast to

the real values of 𝑘𝑎, they increase with increasing frequency (Fig. 1.7, right).

- 24 -

The essential difference between the imaginary components of the eigenvalues 𝑘𝑏

and the corresponding values for ideally conducting parallel planes is explained by the

magnitude of the parameter 𝑑𝑏 (1.34): its smallness is ensured at very low relative

thicknesses of the metallic layer �̃� (�̃� ≪ 1)or at sufficiently high frequencies (�̃� ≫ �̃�𝑟𝑒𝑧).

1.5 Summary

The first chapter establishes the connection between the radiation of a charged

particle and the synchronous electromagnetic eigenmodes of two-layer cylindrical and flat

metal structures.

For a two-layered metal cylindrical waveguide, it is shown, that a single slowly-

propagated (with phase velocity equal to the speed of light) 𝑇𝑀01 mode at certain

frequency (resonant frequency 𝑘 = √1 𝑟0𝑑⁄ , 𝑟0 inner radius of waveguide, 𝑑 inner layer

thickness) is in phase with a radiation field of a charged particle and synchronized with the

ultra-relativistic particle, moving along the waveguide axis. This circumstance is clearly

demonstrated in Figure 1.2, where only one dispersion curve corresponding to the above-

mentioned mode intersects a line describing the phase velocity equal to the speed of light.

The important results of this Chapter also include the calculation of the attenuation

coefficient of the synchronous mode for the resonance frequency (Eq. 1.8), which is

defined as the real component of its phase.

The remarkable properties of the phase of the synchronous mode, presented on

Figure 1.3, are also determined.

The second main direction of this chapter is to investigate the mechanism of

synchronization of the electromagnetic eigen-oscillations of a two-layer flat metallic

structure.

Equations for the eigenvalue numbers of free vibrations in the structure under

consideration are obtained. A subsystem of free oscillations, synchronous with a particle,

moving along a rectilinear trajectory parallel to the symmetry plane of structure, is

distinguished from the complete set of free oscillations. Equations for eigenvalues are

obtained both for a complete system of free oscillations and for their synchronous

- 25 -

subsystem. A unique relationship is established between the roots of the equations for

synchronous oscillations and the complex poles of the integrand for the longitudinal

impedance. Based on graphic constructions, the regularities of the behavior of eigenvalues

are analyzed.

Important difference between the mechanisms of formation of synchronous

oscillations for two-layered cylindrical and planar structures should be mentioned. In the

first case, we have a single synchronous mode, with synchronization being achieved only

at the resonant frequency 𝑘 = √2 𝑟0𝑑⁄ . In the second case, there is an infinite set of eigen-

oscillations, synchronous on the entire infinite frequency range. The radiation field of a

particle is formed by the total contribution of the entire infinite sequence of natural

synchronous oscillations of the structure.

As will be shown in the Second Chapter, the result of the total contribution of

synchronous oscillations is radiation with a narrow-band frequency spectrum and a single

resonant frequency 𝑘 = √1 𝑎𝑑⁄

- 26 -

Chapter 2: Accelerating structures with flat geometry

2.1 Introduction

Impedances and wake functions in multilayer cylindrical structure are studied in [55].

It is shown that under certain conditions there is a narrow-band longitudinal impedance in

the structure. The study of dispersive relations of a two-layer cylindrical structure shows

that in the same conditions the structure becomes a high frequency single slowly (𝑣 < 𝑐)

traveling wave structure. Single-mode structures are good candidates both for novel

accelerating structures and as a source of high frequency monochromatic radiation. As it

is shown in previous chapter, a single-mode structure can be used also for bunch

compression and for microbunching. To be a high frequency single-mode structure, it

requires a high accuracy 𝜇𝑚 – sub-𝜇𝑚 low conductivity internal coating, and an inner

diameter of order of several 𝑚𝑚, which is technically complex problem for structures with

𝑚𝑚 diameters.

In this chapter multilayer flat metallic structure is studied as an alternative of the

abovementioned cylindrical structure. Longitudinal impedance and dispersive relations of

the structure are presented, wake fields generated by a particle travelling through the

structure are observed. Analysis of symmetrical and asymmetrical multilayer flat

waveguides are performed. Conditions in which the longitudinal impedance has a narrow-

band resonance are obtained. Comparison of properties of multilayer flat metallic structure

with cylindrical structure are done.

To calculate electromagnetic fields the field matching technique that implies the

continuity of the tangential components of electric and magnetic fields at the borders of

the layers is used for solving Maxwell equation. The matrix formalism described in [55] is

used to couple electromagnetic field tangential components in two borders of a layer.

Longitudinal impedance of the structure and wake fields in it are obtained via Fourier

inverse transformation of electromagnetic fields.

- 27 -

2.2 Multilayer flat structure

We will start with the most general case: structure consists of two parallel multilayer

infinite plates (Figure 2.1). The boundaries between layers are infinite parallel planes. The

number of layers in the top 𝑛(𝑡) and bottom 𝑛(𝑏) plates can generally be unequal (𝑛(𝑡) ≠

𝑛(𝑏)). The filling of layers may be both dielectric or metallic and in general is characterized

by a static complex dielectric permittivity 휀𝑖(𝑢), 𝑖 = 1,2, . . 𝑛(𝑢), 𝑢 = 𝑡, 𝑏, magnetic permeability

𝜇𝑖(𝑢)

and conductivity 𝜎𝑖(𝑢)

. The dielectric constant material layer can be written as 휀�̃�(𝑢) =

휀𝑖(𝑢) + 𝑗 𝜎𝑖

(𝑢) 𝜔⁄ , where 𝜔 is frequency. For the majority of highly conductive metal generally

accepted 휀𝑖(𝑢) = 휀0; for non-magnetic materials 𝜇𝑖

(𝑢) = 𝜇0 (휀0, 𝜇0 permittivity and magnetic

permeability of vacuum). The thickness of 𝑖-th layer is 𝑑𝑖(𝑢)

. The distance from the plane of

symmetry of the vacuum chamber to the boundary planes, separating layers are equal to

𝑎𝑖(𝑢)

; 𝑎𝑖+1(𝑢) = 𝑎𝑖

(𝑢) + 𝑑𝑖(𝑢), 𝑖 = 1,2,3, …𝑛(𝑢) (for upper half-space).The upper and lower

unbounded regions can be filled by an arbitrary medium, electromagnetic properties of

which are characterized by parameters 휀�̃�(𝑢)+1

(𝑢)= 휀

𝑛(𝑢)+1

(𝑢)+ 𝑗 𝜎

𝑛(𝑢)+1

(𝑢)𝜔⁄ , 𝜇

𝑛(𝑢)+1

(𝑢)= 𝜇0.

The point-like charge is traveling through layered metallic tube with velocity 𝒗

parallel to the 𝑧-axis at vertical offset Δ.

Figure 2.1: Geometry of multilayer flat metallic structure.

- 28 -

Maxwell’s equations must be solved to calculate EM fields excited by the point

charge. Cartesian coordinates 𝑟 = (𝑥, 𝑦, 𝑧) are used, where 𝑦 is a direction of vertical axis.

For (𝑛(𝑡), 𝑛(𝑏)) layered tube there are 𝑛(𝑡) + 𝑛(𝑏) + 3 regions (including inner region and

two outer regions of the structure) and 𝑛(𝑡) + 𝑛(𝑏) + 2 borders. The complete number of

matching equations is 4(𝑛(𝑡) + 𝑛(𝑏) + 2) with 4(𝑛(𝑡) + 𝑛(𝑏) + 2) unknown coefficients as in

each border there are four boundary conditions that match tangential (𝐸𝑥, 𝐸𝑧 , 𝐵𝑥, 𝐵𝑧)

components of electromagnetic fields.

Radiation fields are searched by the method of partial areas [64]. The geometry of

the problem divides into three regions where the fields are presented in different ways:

charge existing region −𝑎−1 ≤ 𝑦 ≤ 𝑎1, metallic layers of the structure −𝑎−n(𝑏) < 𝑦 < −𝑎−1

and 𝑎1 < 𝑦 < 𝑎n(𝑡) and outer vacuum region 𝑦 < −a−n(𝑏) and 𝑦 > an(𝑡). In all regions the

solution is sought in the form of the general solution of the homogeneous Maxwell

equations except for the area between the plates, which, to the general solution of the

homogeneous Maxwell equations is added a particular solution of inhomogeneous

Maxwell's equations.

Electromagnetic fields and point charge are presented via Fourier transformation as

𝐸(𝑥, 𝑦, 𝑧) = ∫ �̃�(𝑘𝑥, 𝑦, 𝑘𝑧)𝑒𝑖𝑘𝑥𝑥𝑒𝑖𝑘(𝑧−𝑐𝑡)𝑑𝑘𝑥𝑑𝑘𝑧

∞

−∞

𝜌(𝑥, 𝑦, 𝑧) = ∫ �̃�(𝑘𝑥, 𝑦, 𝑘𝑧)𝑒𝑖𝑘𝑥𝑥𝑒𝑖𝑘𝑧𝑧𝑑𝑘𝑥𝑑𝑘𝑧

∞

−∞

(2.1)

where 𝑘 = 𝜔 𝑡⁄ is the wave number.

Boundary conditions states that tangential components of full electric and magnetic

fields should be continues at layers boundaries.

E𝑥𝑖 (±𝑎𝑖) = E𝑥

𝑖+1(±𝑎𝑖)

E𝑧𝑖 (±𝑎𝑖) = E𝑧

𝑖+1(±𝑎𝑖)

B𝑥𝑖 (±𝑎𝑖) = B𝑥

𝑖+1(±𝑎𝑖)

B𝑧𝑖 (±𝑎𝑖) = B𝑧

𝑖+1(±𝑎𝑖)

(2.2)

where + and − signs corresponds for layer boundaries in the upper and lower regions,

respectively.

- 29 -

An electric charge traveling in the structure produces electromagnetic field which

can be presented as a sum of two parts

𝑬𝒕𝒐𝒕 = 𝑬𝒑 + 𝑬, 𝑩𝒕𝒐𝒕 = 𝑩𝒑 +𝑩 (2.3)

where 𝑬𝐩, 𝑩𝐩 are electric and magnetic fields in the case of perfectly conducting plates and

𝑬,𝑩 are solutions of source free Maxwell’s equations with a metallic current 𝒋 = 𝜎𝑬. The

last ones’ dependence on the source particles comes only with boundary conditions.

Firstly electric 𝑬𝐩 and magnetic 𝑩𝐩 fields in the perfectly conducting parallel plates

with a static charge in the structure are calculated, then fields when there is a charge

traveling in the direction of structure axis can be obtained via Lorentz transformation [64].

Maxwell’s equations for the structure with a static charge 𝜌 in it are

∇ ∙ 𝑬𝒑 =𝜌

𝜀0

∇ ∙ 𝑩𝒑 = 0

∇ × 𝑬𝒑 = 0

∇ × 𝑩𝒑 = 0

(2.4)

where ∇ is a nabla operator.

Using Fourier transformation in Maxwell’s equations and expressing electric field 𝑥

and 𝑦 components via 𝑧 component, the wave equation for a point charge can be derived

𝜕2𝐸𝑝𝑧

𝜕𝑦2− 𝛼2𝐸𝑝𝑧 =

𝑞

4𝜋2𝜀0𝑖𝑘𝑧𝛿(𝑦 − Δ) (2.5)

where 𝛼 = √𝑘𝑥2 + 𝑘𝑧

2 and Δ is the point charge deviation from the center of 𝑦 axis.

The solution of Eq. 2.5 is a sum of homogenous equation solution (�̃�𝑧,ℎ) and a

particular solution (�̃�𝑧,𝑝).

Presenting the solution of homogenous part as a sum of hyperbolic cosine and sine

with unknown coefficients and finding a particular solution by integrating two parts of the

wave equation, its full solution can be obtained.

𝐸𝑝𝑧±(𝑘𝑥, 𝑦, 𝑘𝑧) = −

𝑞𝑖𝑘𝑧

4𝜋2𝛼𝜀0

sinh(𝛼(𝑎±Δ)) sinh(𝛼(𝑎∓𝑦))

sinh(2𝛼𝑎) (2.6)

- 30 -

where + sign stand for an electric field in upper part of the structure (𝑦 > Δ ) and – for an

electric field in lower part (𝑦 < Δ ).

𝑥 and 𝑦 component of the electric field can be obtained via Faraday’s low and

magnetic field is zero for a static charge.

𝐸𝑝𝑥±(𝑘𝑥, 𝑦, 𝑘𝑧) = −

𝑞𝑖𝑘𝑥

4𝜋2𝛼𝜀0

𝑠𝑖𝑛ℎ(𝛼(𝑎∓𝑦)) 𝑠𝑖𝑛ℎ(𝛼(𝑎±𝛥))

𝑠𝑖𝑛ℎ(2𝛼𝑎)

𝐸𝑝𝑦±(𝑘𝑥, 𝑦, 𝑘𝑧) = ±

𝑞

4𝜋2𝜀0

𝑐𝑜𝑠(𝛼(𝑎∓𝑦)) 𝑠𝑖𝑛ℎ(𝛼(𝑎±𝛥))

𝑠𝑖𝑛ℎ(2𝛼𝑎)

(2.7)

The Fourier components of the electromagnetic fields of a moving charge can be

obtained via Lorentz transformation

𝐸𝑝𝑧±(𝑘𝑥, 𝑦, 𝑘𝑧) = −

1

𝛾2𝑞𝑗𝑘𝑧

4𝜋2𝑘𝑥𝜀0

sinh(𝑘𝑥(𝑎±Δ)) sinh(𝑘𝑥(𝑎∓𝑦))

sinh(2𝑘𝑥𝑎)

𝐸𝑝𝑥±(𝑘𝑥, 𝑦, 𝑘𝑧) = 𝛾

2 𝑘𝑥

𝑘𝑧𝐸𝑝𝑧± (𝑘𝑥, 𝑦, 𝑘𝑧)

𝐸𝑝𝑦±(𝑘𝑥, 𝑦, 𝑘𝑧) = ±𝛾

2𝑗𝑘𝑥

𝑘𝑧coth(𝑘𝑥(𝑎 ∓ 𝑦))𝐸𝑝𝑧

± (𝑘𝑥, 𝑦, 𝑘𝑧)

𝐵𝑝𝑧±(𝑥, 𝑦, 𝑧) = 0

𝐵𝑝𝑥±(𝑘𝑥, 𝑦, 𝑘𝑧) =

𝑣

𝑐2𝐸𝑝𝑦± (𝑘𝑥, 𝑦, 𝑘𝑧)

𝐵𝑝𝑦±(𝑘𝑥, 𝑦, 𝑘𝑧) = −

𝑣

𝑐2𝐸𝑝𝑥± (𝑘𝑥, 𝑦, 𝑘𝑧)

(2.8)

note that variable of integration is replaced by 𝑘𝑧 → 𝑘𝑧′ 𝛾⁄ and 𝛼 = √𝑘𝑥

2 + (𝑘𝑧′

𝛾)2

≈ 𝑘𝑥.

We write fields in Eq. 2.8 in the form of plane waves

𝐸𝑝𝑥,𝑦,𝑧 = 𝐸1,𝑝𝑥,𝑦,𝑧 + 𝐸2,𝑝𝑥,𝑦,𝑧 = 𝐴1,𝑥,𝑦,𝑧(±)

𝑒(∓)𝑘𝑎𝑒−𝑘𝑦 + 𝐴2,𝑥,𝑦,𝑧(±)

𝑒(±)𝑘𝑎𝑒−𝑘𝑦

𝐵𝑝𝑥,𝑦,𝑧 = 𝐵1,𝑝𝑥,𝑦,𝑧 + 𝐵2,𝑝𝑥,𝑦,𝑧 = 𝐵1,𝑥,𝑦,𝑧(±)

𝑒(∓)𝑘𝑎𝑒−𝑘𝑦 + 𝐵2,𝑥,𝑦,𝑧(±)

𝑒(±)𝑘𝑎𝑒−𝑘𝑦 (2.9)

where

𝐴1,𝑧(±)= −𝑗𝛾−2

𝑞𝑘𝑧

8𝜋2𝜀0𝑘cosh(2𝑘𝑎) sinh(𝑘(𝑎 ± ∆)) 𝐴2,𝑧

(±)= −𝐴1,,𝑧

(±)

𝐴1,𝑥(±)= −𝑗

𝑞𝑘𝑥

8𝜋2𝜀0𝑘𝐶𝑠ℎ(2𝑘𝑎)𝑆ℎ{𝑘(𝑎 ± ∆)} = 𝛾2𝐴1,𝑧

(±)𝐴2,𝑧(±)= −𝐴2,𝑧

(±)

𝐴1,𝑦(±)= ±

𝑞

8𝜋2𝜀0𝐶𝑠ℎ(2𝑘𝑎)𝑆ℎ{𝑘(𝑎 ± ∆)} 𝐴2,𝑦

(±)= 𝐴2,,𝑦

(±)

(2.10)

On the metallic surfaces the electromagnetic field components are

- 31 -

𝐸𝑝𝑥(𝑘𝑥, ±𝑎, 𝑘𝑧) = 𝐸𝑝𝑧

(𝑘𝑥, ±𝑎, 𝑘𝑧) = 0

𝐸𝑝𝑦(𝑘𝑥, ±𝑎, 𝑘𝑧) =

𝑞

4𝜋2𝜀0

sinh(𝑘𝑥(Δ±𝑎))

sinh(2𝑘𝑥𝑎)

𝐵𝑝𝑥(𝑘𝑥, ±𝑎, 𝑘𝑧) =

𝑞

4𝜋2𝜀0

𝑣

𝑐2

sinh(𝑘𝑥(Δ±𝑎))

sinh(2𝑘𝑥𝑎)

𝐵𝑝𝑦(𝑘𝑥, ±𝑎, 𝑘𝑧) = 𝐵𝑝𝑧

(𝑘𝑥, ±𝑎, 𝑘𝑧) = 0

(2.11)

The only non-zero tangential component at the boundaries 𝑦 = ±𝑎 is a component

𝐵𝑝𝑥, which in ultrarelativistic approximation (𝑣 → 𝑐) is

𝐵𝑝𝑥(𝑘𝑥, ±𝑎, 𝑘𝑧) =

𝑞

4𝜋2𝜀0𝑐

sinh(𝑘𝑥(Δ±𝑎))

sinh(2𝑘𝑥𝑎) (2.12)

As it was mentioned, the total electromagnetic field excited by a point charge

traveling in the multilayer structure was presented as a sum of two parts. Now the second

part of total electromagnetic field will be calculated, i.e. solutions of the source free

Maxwell’s equations with a metallic current 𝒋 = 𝜎𝑬. The solution inside layers is sought in

the form of the general solution of the homogenous Maxwell’s equations.

𝑬(𝑖)(𝑥, 𝑦, 𝑧) = 𝑬1(𝑖)(𝑥, 𝑦, 𝑧) + 𝑬2

(𝑖)(𝑥, 𝑦, 𝑧)

𝑩(𝑖)(𝑥, 𝑦, 𝑧) = 𝑩1(𝑖)(𝑥, 𝑦, 𝑧) + 𝑩2

(𝑖)(𝑥, 𝑦, 𝑧) (2.13)

where

𝑬1,2(𝑖)(𝑥, 𝑦, 𝑧) = ∫ ∫ 𝑬1,2

(𝑖)(𝑘𝑥, 𝑦, 𝑘𝑧)𝑒𝑗(𝑘𝑥𝑥+𝑘𝑧(𝑧−𝑣𝑡))

∞

−∞𝑑𝑘𝑥𝑑𝑘

∞

−∞

𝑩1,2(𝑖)(𝑥, 𝑦, 𝑧) = ∫ ∫ 𝑩1,2

(𝑖)(𝑘𝑥, 𝑦, 𝑘𝑧)𝑒𝑗(𝑘𝑥𝑥+𝑘𝑧(𝑧−𝑣𝑡))

∞

−∞𝑑𝑘𝑥𝑑𝑘

∞

−∞

(2.14)

with

𝑬1,2(𝑖)(𝑘𝑥, 𝑦, 𝑘𝑧) = 𝑨1,2

(𝑖)𝑒∓𝑘𝑦

(𝑖)𝑦

𝑩1,2(𝑖)(𝑘𝑥, 𝑦, 𝑘𝑧) =

𝑒−𝑗(𝑘𝑥𝑥+𝑘𝑧(𝑧−𝑣𝑡))

𝑗𝑘𝑧𝑣𝑟𝑜𝑡 [𝑬1,2

(𝑖)(𝑘𝑥, 𝑦, 𝑘𝑧)𝑒𝑗(𝑘𝑥𝑥+𝑘𝑧(𝑧−𝑣𝑡))]

(2.15)

For the inner region (−𝑎−1 < 𝑦 < 𝑎1) we should add the partial solution (𝑬𝑝, 𝑩𝑝) of

inhomogeneous Maxwell’s equation (Eq. 2.8) to the general solution of homogeneous

Maxwell’s equation

- 32 -

𝑬(0)(𝑥, 𝑦, 𝑧) = 𝑬1(𝑖)(𝑥, 𝑦, 𝑧) + 𝑬2

(𝑖)(𝑥, 𝑦, 𝑧) + 𝑬𝑝(𝑥, 𝑦, 𝑧)

𝑩(0)(𝑥, 𝑦, 𝑧) = 𝑩1(𝑖)(𝑥, 𝑦, 𝑧) + 𝑩2

(𝑖)(𝑥, 𝑦, 𝑧) + 𝑩𝑝(𝑥, 𝑦, 𝑧) (2.16)

Finally, for the upper and lower outer regions fields are sought in the form of

vanishing at infinity (at 𝑦 → ±∞)

𝑬(𝑛(𝑡)+1)(𝑥, 𝑦, 𝑧) = 𝑬1

(𝑛(𝑡)+1)(𝑥, 𝑦, 𝑧)

𝑩(𝑛(𝑡)+1)(𝑥, 𝑦, 𝑧) = 𝑩1

(𝑛(𝑡)+1)(𝑥, 𝑦, 𝑧)

𝑬(−𝑛(𝑏)−1)(𝑥, 𝑦, 𝑧) = 𝑬2

(−𝑛(𝑏)−1)(𝑥, 𝑦, 𝑧)

𝑩(−𝑛(𝑏)−1)(𝑥, 𝑦, 𝑧) = 𝑩2

(−𝑛(𝑏)−1)(𝑥, 𝑦, 𝑧)

(2.17)

for top (𝑦 > 𝑎𝑛(𝑡)+1) and bottom (𝑦 < −𝑎−𝑛(𝑏)−1) regions, respectively.

As it is mentioned, we obtain a system of 4(𝑁 +𝑀 + 2) equations for the same

number of unknown amplitudes. Amplitudes 𝐴𝑧,1𝑖 and 𝐴𝑧,2

𝑖 can be presented by 𝐴𝑥,1𝑖 , 𝐴𝑦,1

𝑖

and 𝐴𝑥,2𝑖 , 𝐴𝑦,2

𝑖 𝐴𝑥,1 via Gauss’s low

𝐴𝑧,1𝑖 = −

𝑘𝑥𝐴𝑥,1𝑖 +𝑖𝑘𝑦

𝑖 𝐴𝑦,1𝑖

𝑘𝑧

𝐴𝑧,2𝑖 = −

𝑘𝑥𝐴𝑥,2𝑖 −𝑖𝑘𝑦

𝑖 𝐴𝑦,2𝑖

𝑘𝑧

(2.18)

and magnetic field components can be derived via Faraday’s low.

To determine the electromagnetic field we will use the field matching technique that

implies the continuity of the tangential components of electromagnetic fields at the borders

of layers (𝑦 = ±𝑎𝑖). We are interested in tangential components of fields, so we will

introduce the tangential field vector �̂�𝑖(𝐸𝑥𝑖, 𝐸𝑧

𝑖 , c𝐵𝑥𝑖 , c𝐵𝑧

𝑖). We will use the matrix approach

introduced in [63] to obtain electromagnetic fields in the vacuum region.

Connection between the components of the fields and amplitudes is realized by

using the basic transformation matrixes

- 33 -

�̂�1(𝑖)(𝑦) =

𝑒−𝑘𝑦𝑖 𝑦

𝑘𝑧2

(

𝑘𝑧2 0 0 0

−𝑘𝑥𝑘𝑧 −𝑖𝑘𝑦𝑖 𝑘𝑧 0 0

−𝑖𝑘𝑦𝑖 𝑘𝑥 𝑘𝑦

𝑖 2 − 𝑘𝑧2 0 0

−𝑖𝑘𝑦𝑖 𝑘𝑧 𝑘𝑥𝑘𝑧 0 0)

(2.19)

�̂�2(𝑖)(𝑦) =

𝑒𝑘𝑦𝑖 𝑦

𝑘𝑧2

(

0 0 𝑘𝑧2 0

0 0 −𝑘𝑥𝑘𝑧 −𝑖𝑘𝑦𝑖 𝑘𝑧

0 0 𝑖𝑘𝑦𝑖 𝑘𝑥 𝑘𝑦

𝑖 2 − 𝑘𝑧2

0 0 −𝑖𝑘𝑦𝑖 𝑘𝑧 𝑘𝑥𝑘𝑧 )

(2.20)

�̂�(𝑖)(𝑦) = �̂�1(𝑖)(𝑦)+�̂�2

(𝑖)(𝑦) (2.21)

where 𝑘𝑦(𝑖) = 𝑘𝑥

2 + (1 − 휀(𝑖)𝜇(𝑖)𝑣2)𝑘𝑧2.

Let's establish a connection between the values of the fields on the lower borders of

the neighboring layers. The fields on the upper and lower borders of the 𝑖-th layer of the

upper half-space can be expressed in terms of the values of the same amplitudes:

�̂�𝑖(𝑎𝑖) = �̂�1𝑖(𝑎𝑖)�̂�

𝑖

�̂�𝑖(𝑎𝑖+1) = �̂�1𝑖(𝑎𝑖+1)�̂�

𝑖⇒ �̂�𝑖(𝑎𝑖+1) = �̂�1

𝑖(𝑎𝑖+1)�̂�1𝑖(𝑎𝑖)

−1�̂�𝑖(𝑎𝑖) (2.22)

From field tangential components equality on layer borders (�̂�𝑖+1(𝑎𝑖+1) = �̂�𝑖(𝑎𝑖+1))

one obtains

�̂�𝑖+1(𝑎𝑖+1) = �̂�1𝑖(𝑎𝑖+1)�̂�1

𝑖(𝑎𝑖)−1�̂�𝑖(𝑎𝑖) (2.23)

Consistent application of (2.23) for upper half-spaces gives

�̂�𝑛(𝑡)+1(𝑎𝑛(𝑡)+1) = 𝑄

(𝑡)�̂�0(𝑎1) (2.24)

where 𝑄(𝑡) = ∏ 𝑄𝑖(𝑡)𝑛(𝑡)

𝑖=1 and 𝑄𝑖(𝑡) = �̂�(𝑖)(𝑎𝑖+1)�̂�

(𝑖)(𝑎𝑖)−1.

Similar relation can be obtained for lower half-space

�̂�(−(𝑛(𝑏)+1))

(−𝑎−(𝑛(𝑏)+1)) = 𝑄(𝑏)�̂�(0)(−𝑎−1) (2.25)

where 𝑄(𝑏) = ∏ 𝑄𝑖(𝑏)𝑛(𝑏)

𝑖=1 , 𝑄𝑖(𝑏) = �̂�(−𝑖)(−𝑎−(𝑖+1))�̂�

(−𝑖)(−𝑎−𝑖)−1.

- 34 -

For the symmetrical case 𝑄𝑖(𝑏) = 𝑄𝑖

(𝑡)−1 and, respectively, 𝑄(𝑏) = ∏ 𝑄𝑖

(𝑡)−1𝑛𝑖=1 .

Given the matching conditions on the borders of the plates, we have finally

�̂�1(𝑛(𝑡)+1)

(𝑎𝑛(𝑡)+1)�̂�(𝑡𝑏) = 𝑄𝑡[�̂�(0)(𝑎1)�̂�

(0) + �̂�(𝑡)] (2.26)

�̂�2(−(𝑛(𝑏)+1))

(−𝑎−(𝑛(𝑏)+1)) �̂�(𝑡𝑏) = 𝑄𝑏[�̂�(0)(−𝑎−1)�̂�

(0) + �̂�(𝑏)] (2.27)

where �̂�(𝑡,𝑏) and �̂�(𝑡,𝑏) are the single-column matrixes

�̂�(𝑡,𝑏) = {{0}, {0}, {2𝑘𝑦(0)𝑣 csch (2𝑘𝑦

(0)𝑎) sinh [𝑘𝑦

(0)(∆ ± 𝑎1)]} , {0}}

�̂�(𝑡𝑏) = {{𝐴𝑥,1(𝑛(𝑡)+1)

} , {𝐴𝑦,1(𝑛(𝑡)+1)

} , {𝐴𝑥,2(−(𝑛(𝑏)+1))

} , {𝐴𝑦,2(−(𝑛(𝑏)+1))

}} (2.28)

On the Eq. 2.9 and Eq. 2.10 condition that fields are vanishing on 𝑦 = ±∞ is taken

into account.

The amplitudes �̂�(0) may be expressed in terms of amplitudes �̂�(𝑡𝑏) using Eq. 2.26

and Eq. 2.27

�̂�(0) = �̂�𝑡�̂�(𝑡𝑏) − �̂�(0)(𝑎1)−1�̂�(𝑡)

�̂�(0) = �̂�𝑏�̂�(𝑡𝑏) − �̂�(0)(−𝑎−1)−1�̂�(𝑏)

(2.29)

where

�̂�𝑡 = �̂�(0)(𝑎1)−1𝑇𝑡

−1�̂�1(𝑛(𝑡)+1)

(𝑎𝑛(𝑡)+1)

�̂�𝑏 = �̂�(0)(−𝑎−1)−1𝑇𝑏

−1�̂�2(−(𝑛(𝑏)+1))

(−𝑎−(𝑛(𝑏)+1)) (2.30)

Eliminating the outer layer coefficients we obtain explicit expressions for the

amplitudes of the radiation field of the particle in the space between the plates

�̂�(0) =1

2(�̂�(+)�̂�(−)

−1�̂�(−) − �̂�(+)) (2.31)

with

�̂�(±) = �̂�𝑡 ± �̂�𝑏, �̂�(±) = �̂�(0)(𝑎1)−1�̂�(𝑡) ± �̂�(0)(−𝑎−1)

−1�̂�(𝑏) (2.32)

- 35 -

Amplitudes of the radiation field for the structure with an arbitrary number of layers

in each of the plates can be obtained by Eq. 2.31 numerically. The particle velocity can

also be arbitrary.

2.3 Two-layer flat structure

Note that in the symmetric case (𝑎𝑖 = 𝑎−𝑖, 𝑘𝑦(𝑖) = 𝑘𝑦

(−𝑖), 𝑛(𝑡) = 𝑛(𝑏)) 𝑈𝑖(𝑏) = 𝑈𝑖

(𝑡)−1 and

𝑄(𝑏) = 𝑄(𝑡)−1

. For some important special cases, it is possible to obtain explicit

expressions.

Figure 2.2: Two-layer flat structure. Geometry and electromagnetic parameters.

An important special case is a symmetrical two-layer structure with unbounded

external walls (Figure 2.2). We write the amplitudes of the longitudinal electric component

of the radiation field of the particles in the vacuum region between the plates (corresponds

to Eq. 2.15 for 𝑖 = 0)

𝐴𝑧,1,2(0)

= 𝐴𝐶 cosh (𝑘𝑦(0)Δ) ∓ 𝐴𝑆 sinh (𝑘𝑦

(0)Δ) (2.33)

where

𝐴𝐶 = 𝑗𝑞𝑘𝑧

16𝜋2𝜀0

𝑣2

𝑐2 𝐴+𝐵 tanh(𝑘𝑦

(0)𝑎1)

𝐷1𝐶𝐷2

𝐶 , 𝐴𝑆 = 𝑗𝑞𝑘𝑧

16𝜋2𝜀0

𝑣2

𝑐2 𝐴+𝐵 coth(𝑘𝑦

(0)𝑎1)

𝐷1𝑆𝐷2

𝑆 (2.34)

The following notations are made in Eq. 2.34

𝐴 = 𝑘𝑦(0)[(𝑘𝑦

(1)2− 𝑘𝑦

(2)2) (𝑘𝑥

2𝐾12 − 𝑘𝑦

(1)2𝑘𝑧2) + (𝑘𝑦

(1)2− 𝑘𝑥

2) 𝑃]

휀1, 𝜎1

휀1, 𝜎1

휀2, 𝜎2

휀2, 𝜎2

- 36 -

𝐵 = 2𝑘𝑦(1)(𝑘𝑥

2 − 𝑘𝑦(0)2) 𝑆1�̃�2

𝐷1𝐶 = 𝑘𝑦

(1)𝑆2 cosh (𝑘𝑦

(0)𝑎1) + 𝑘𝑦

(0)𝑆1 sinh (𝑘𝑦

(0)𝑎1)

𝐷1𝑆 = 𝑘𝑦

(1)𝑆2 sinh (𝑘𝑦

(0)𝑎1) + 𝑘𝑦

(0)𝑆1 cosh (𝑘𝑦

(0)𝑎1)

𝐷2𝐶 = 𝑘𝑦

(0)𝐾12�̃�1 cosh (𝑘𝑦

(0)𝑎1) + 𝑘𝑦

(1)𝐾02�̃�2 sinh (𝑘𝑦

(0)𝑎1)

𝐷2𝑆 = 𝑘𝑦

(0)𝐾12�̃�1 sinh (𝑘𝑦

(0)𝑎1) + 𝑘𝑦

(1)𝐾02�̃�2 cosh (𝑘𝑦

(0)𝑎1) (2.35)

𝑆1 = 𝑘𝑦(1) cosh(𝑘𝑦

(1)𝑑) + 𝑘𝑦(2) sinh(𝑘𝑦

(1)𝑑) 𝑆2 = 𝑘𝑦(2) cosh(𝑘𝑦

(1)𝑑) + 𝑘𝑦(1) sinh(𝑘𝑦

(1)𝑑)

�̃�1 = 𝑘𝑦(1)𝐾2

2 cosh(𝑘𝑦(1)𝑑) + 𝑘𝑦

(2)𝐾12 sinh(𝑘𝑦

(1)𝑑) �̃�2 = 𝑘𝑦(2)𝐾1

2 cosh(𝑘𝑦(1)𝑑) + 𝑘𝑦

(1)𝐾22 sinh(𝑘𝑦

(1)𝑑)

𝑃 = (𝑘𝑦(2)2𝐾1

2 + 𝑘𝑦(1)2𝐾2

2) cosh(2𝑘𝑦(1)𝑑) + 𝑘𝑦

(1)𝑘𝑦(2)(𝐾1

2 + 𝐾22) sinh(2𝑘𝑦

(1)𝑑)

𝐾𝑖2 = 𝑘𝑦

(𝑖)2− 𝑘𝑥

2 − 𝑘𝑧2, 𝑖 = 0,1,2

Result of Eq. 2.33 – Eq. 2.35 is a generalization of the well-known result [64] on the

two-layer structure for an arbitrary velocity of the particle. In the transition to single-layer

unbounded plates (𝑘𝑦(1) = 𝑘𝑦

(2)) the first term in 𝐴 vanishes. In the transition to ultra-

relativistic approximation (𝑘𝑦(0) = 𝑘𝑥) factor 𝐵 in vanishes. The following are the values of

the field amplitudes for the structure, consisting of two unbounded parallel plates and the

arbitrary particle velocity.

𝐴𝐶 = 𝑗𝑞𝑘𝑧

8𝜋2𝜀0

𝑣2

𝑐2

(𝑘𝑥2−𝑘𝑦

(1)2)𝑘𝑦(0)+𝑘𝑦

(1)(𝑘𝑥2−𝑘𝑦

(0)2) tanh(𝑘𝑦

(0)𝑎1)

(𝑘𝑦(1)cosh(𝑘𝑦

(0)𝑎1)+𝑘𝑦

(0)sinh(𝑘𝑦

(0)𝑎1))(𝑘𝑦

(0)𝐾12 cosh(𝑘𝑦

(0)𝑎1)+𝑘𝑦

(1)𝐾02 sinh(𝑘𝑦

(0)𝑎1))

𝐴𝑆 = 𝑗𝑞𝑘𝑧

8𝜋2𝜀0

𝑣2

𝑐2

(𝑘𝑥2−𝑘𝑦

(1)2)𝑘𝑦(0)+𝑘𝑦

(1)(𝑘𝑥2−𝑘𝑦

(0)2) coth(𝑘𝑦

(0)𝑎1)

(𝑘𝑦(0)cosh(𝑘𝑦

(0)𝑎1)+𝑘𝑦

(1)sinh(𝑘𝑦

(0)𝑎1))(𝑘𝑦

(1)𝐾02 cosh(𝑘𝑦

(0)𝑎1)+𝑘𝑦

(0)𝐾12 sinh(𝑘𝑦

(0)𝑎1))

(2.36)

At 𝑘𝑦(0) = 𝑘𝑥 and 𝑣 = 𝑐 amplitudes Eq. 2.36 coincide with the corresponding

amplitudes in [64], obtained in the ultra-relativistic approximation.

- 37 -

In the case of the high conductivity of outer layer, it can be considered as a perfect

conductor. Transition to this case is made by means of equating to infinity in Eq. 2.34 the

wave number 𝑘𝑦(2)

. In this case the field amplitudes are

𝐴1𝐶 = 𝑗𝑞𝑘𝑧

16𝜋2𝜀0

𝑣2

𝑐2

(𝑘𝑥2−𝑘𝑦

(1)2)𝑘𝑦(0)sinh(2 𝑘𝑦

(1)𝑑)+2𝑘𝑦

(1)(𝑘𝑦(0)2

−𝑘𝑥2) sinh2(𝑘𝑦

(1)𝑑) tanh(𝑘𝑦

(0)𝑎1)

𝑅1𝐶𝑅2

𝐶

𝐴1𝑆 = 𝑗𝑞𝑘𝑧

16𝜋2𝜀0

𝑣2

𝑐2

(𝑘𝑥2−𝑘𝑦

(1)2)𝑘𝑦(0)sinh(2 𝑘𝑦

(1)𝑑)+2𝑘𝑦

(1)(𝑘𝑦(0)2

−𝑘𝑥2) sinh2(𝑘𝑦

(1)𝑑) coth(𝑘𝑦

(0)𝑎1)

𝑅1𝑆𝑅2𝑆

(2.37)

where the following notations are made

𝑅1𝐶 = 𝑘𝑦

(1) cosh(𝑘𝑦(1)𝑑) cosh(𝑘𝑦

(0)𝑎1) + 𝑘𝑦(0) sinh(𝑘𝑦

(1)𝑑) sinh(𝑘𝑦(0)𝑎1)

𝑅1𝑆 = 𝑘𝑦

(1)cosh(𝑘𝑦

(1)𝑑) sinh(𝑘𝑦

(0)𝑎1) + 𝑘𝑦

(0)sinh(𝑘𝑦

(1)𝑑) cosh(𝑘𝑦

(0)𝑎1)

𝑅2𝐶 = 𝑘𝑦

(0)𝐾12 cosh(𝑘𝑦

(1)𝑑) cosh(𝑘𝑦(0)𝑎1) + 𝑘𝑦

(1)𝐾02 sinh(𝑘𝑦

(1)𝑑) sinh(𝑘𝑦(0)𝑎1)

𝑅2𝑆 = 𝑘𝑦

(0)𝐾12 cosh(𝑘𝑦

(1)𝑑) sinh(𝑘𝑦(0)𝑎1) + 𝑘𝑦

(1)𝐾02 sinh(𝑘𝑦

(1)𝑑) cosh(𝑘𝑦(0)𝑎1)

(2.38)

Below the longitudinal electric field in case of asymmetric filling of plates is

presented. A structure with perfectly conducting plates, and only one of them covered with

a layer of finite conductivity material is considered.

𝐸𝑧 = 𝑗

𝑞𝑘𝑧

16𝜋2𝜀0{𝐴1𝑧𝑒

−𝑘𝑥𝑎1 + 𝐴2𝑧𝑒𝑘𝑥𝑎1} = 𝑗

𝑞𝑘𝑧𝐴0

8𝜋2𝜀0𝐶ℎ(2𝑘𝑥𝑎1)

𝐴1,2𝑧 = −𝐴0𝑆ℎ{𝑘𝑥(𝑎 + ∆)}𝑒∓𝑘𝑥𝑎1

(2.39)

with

𝐴0 = 𝑘𝑥 (𝑘𝑦(1)2 − 𝑘𝑥

2) 𝑆ℎ(2𝑑𝑘y(1)) {(𝑘𝑥𝐶ℎ(2𝑎1𝑘𝑥)𝑆ℎ(𝑑𝑘𝑦

(1)) + 𝑘𝑦(1)𝑆ℎ(2𝑎1𝑘𝑥)𝐶ℎ(𝑑𝑘𝑦

(1)))

(𝑘𝑦(1)𝑘𝑧

2𝐶ℎ(2𝑎1𝑘𝑥)𝑆ℎ(𝑑𝑘𝑦(1)) − 𝑘𝑥𝐾1

2𝑆ℎ(2𝑎1𝑘𝑥)𝐶ℎ(𝑑𝑘𝑦(1)))}

(2.40)

2.4 Impedance of two-layer symmetrical flat structure

As it is mentioned in the first chapter, the longitudinal impedance is a Fourier

transformation of normalized longitudinal component of Lorentz force. General expression

of Lorentz force is

𝑭(𝑥, 𝑦, 𝑧) = 𝑞[𝑬(𝑥, 𝑦, 𝑧) + 𝒗 × 𝑩(𝑥, 𝑦, 𝑧)] (2.41)

- 38 -

The longitudinal (𝑧 direction) impedance 𝑍∥(𝑥, 𝑦, 𝑘𝑧) is a Fourier transformation of

normalized longitudinal component of Lorenz force 𝐹𝑧(𝑘𝑥, 𝑦, 𝑘𝑧) . In the case, when a

charge 𝑞 is traveling along 𝑧 axis, the longitudinal component of Lorentz force becomes

𝐹𝑧(𝑘𝑥, 𝑦, 𝑘𝑧) = 𝑞𝐸𝑧(𝑘𝑥, 𝑦, 𝑘𝑧) (2.42)

and consequently the longitudinal impedance becomes

𝑍∥(𝑥, 𝑦, 𝑘𝑧) = −1

𝑞𝑐∫ 𝐸𝑧(𝑘𝑥, 𝑦, 𝑘𝑧)𝑒

−𝑖𝑘𝑥𝑥𝑑𝑘𝑥∞

−∞ (2.43)

Consider the most important and at the same time quite simple special case:

symmetrical two-layer flat structure. In the case of the perfectly conductive outer layer

(𝑘𝑦(2) → ∞) and the particle velocity equal to the velocity of light (𝑘𝑦

(0) = 𝑘𝑥, 𝑣 = 𝑐), the

formulae Eq. 2.37 are simplified to the form

𝐴1𝐶(𝑆) = 𝑗𝑞𝑘𝑧𝑘𝑦

(1)

16𝜋2𝜀0

(𝑘𝑥2−𝑘𝑦

(1)2) sinh(2𝑘𝑦

(1)𝑑)

𝑅1𝐶(𝑆)𝑅2

𝐶(𝑆) (2.44)

where

𝑅1𝐶

𝑅2𝐶} = {

𝑘𝑦(1)

𝑘𝑥𝐾12} 𝐶ℎ(𝑘𝑦

(1)𝑑)𝐶ℎ(𝑘𝑥𝑎1) ± {𝑘𝑥

𝑘𝑦(1)𝑘𝑥2} 𝑆ℎ(𝑘𝑦

(1)𝑑)𝑆ℎ(𝑘𝑥𝑎1)

𝑅1𝑆

𝑅2𝑆} = {

𝑘𝑦(1)

𝑘𝑥𝐾12} 𝐶ℎ(𝑘𝑦

(1)𝑑)𝑆ℎ(𝑘𝑥𝑎1) ± {𝑘𝑥

𝑘𝑦(1)𝑘𝑥

2} 𝑆ℎ(𝑘𝑦(1)𝑑)𝐶ℎ(𝑘𝑥𝑎1)

(2.45)

This case is fairly well simulates the highly conducting outer wall and is a two-

dimensional analogue of a two-layer cylindrical tube, the unique resonant properties were

discovered earlier. In the cylindrical waveguide with two-layer metal walls under high

conductivity of the thick outer wall and the relatively low conductivity of the thin inner layer,

the radiation of the driving particles has a narrow-band resonance. Resonance is the sole,

and formed by single slowly propagating waveguide mode. We show that the same is true

of the two-dimensional two-layer metal structure.

In the cylindrical case, the resonance frequency is equal to 𝑘𝑧,𝑟𝑒𝑧 = √2 𝑎1𝑑⁄ (𝑎1

waveguide inner radius, 𝑑 inner layer thickness). Minimum damping (in the case of perfect

- 39 -

conductor outer layer) is provided by the condition 휁 =1

√3𝑍0𝜎1𝑑 (𝑍0 = 120𝜋 Ω, 𝜎1 -inner

layer conductivity).

Let us demonstrate in the first step the single narrow-band extremal nature of the

longitudinal impedance of the two-dimensional two-layer structure.

Figure 2.3: Real (left) and imaginary (right) parts of the longitudinal impedance of two-layer flat

structure. On the graphs, the thicknesses of the inner layer in micrometers are shown.

Figure 2.3 shows a series of distributions of real and imaginary components of the

longitudinal impedance of a two-layer planar structure with a perfectly conductive outer

layer (Figure4). The distance between the plates is 2 𝑐𝑚 (half-height 𝑎1 = 1 𝑐𝑚) and the

conductivity of the inner layer is 2 × 103 Ω−1𝑚−1 The numbers on the curves in the graph

indicate the thickness of the inner layer (𝜇𝑚): "1" (1𝜇𝑚), "2" (2𝜇𝑚) and so on. As we see,

the longitudinal impedance has a single narrow-band resonance with a resonance

frequency dependent on the thickness of the inner layer. As is shown in Table 1, it is quite

well defined by the formula

𝑓𝑟𝑒𝑧 =𝑐

2𝜋√

1

𝑎1𝑑 (2.46)

where 𝑎1 = 𝑎 2⁄ is a half-height of a structure.

- 40 -

𝑑 (𝜇𝑚) 1 2 3 4 5 6 7 8 10 16

𝑓𝑟𝑒𝑧 (𝑇𝐻𝑧) 0.477 0.338 0.276 0.239 0.214 0.195 0.180 0.169 0.151 0.119

𝑓𝑚𝑎𝑥 (𝑇𝐻𝑧) 0.479 0.339 0.277 0.241 0.215 0.197 0.183 0.171 0.154 0.124

𝑓0 (𝑇𝐻𝑧) 0.479 0.503 0.277 0.241 0.215 0.197 0.183 0.171 0.154 0.124

𝜍𝑖 0.251 0.339 0.754 1.001 1.257 1.508 1.760 2.011 2.513 4.021

Table 2.1: Resonance frequencies of two-layer flat structure calculated analytically (𝑓𝑟𝑒𝑧),

numerically (𝑓𝑚𝑎𝑥), impedance imaginary component vanishing frequency (𝑓0) and a

parameter 𝜍𝑖.

In Table 2.1: line 𝑓𝑟𝑒𝑧 shows the resonance frequency calculated by Eq. 2.46; in line

𝑓𝑚𝑎𝑥 the resonance frequencies calculated numerically by the exact formulae are given;

line 𝑓0 shows the frequency values in which the imaginary components of the impedance

vanish; in the last row given the values of the parameter 𝜍𝑖 =1

3𝑍0𝜎1𝑑𝑖. As the table shows,

there is a good coincidence between the frequency values 𝑓𝑟𝑒𝑧 and 𝑓𝑚𝑎𝑥 . Thus, the

resonant frequency in a flat two-layer metal structure, in contrast to the corresponding

cylindrical structure, is defined by Eq. 2.46. In cylindrical case 𝑓𝑟𝑒𝑧 =𝑐

2𝜋√

2

𝑎1𝑑., i.e. in √2

times higher. There is also a complete coincidence of frequencies 𝑓𝑚𝑎𝑥 and 𝑓0, i.e. the

maximum value of the impedance is purely real. Comparing the maximum values of the

real component of impedance distributions (Figure 2.3, left) with the corresponding

parameter 𝜍𝑖 values, we note the following regularities: impedance reaches the highest

value at 𝑑 = 4 𝜇𝑚 , which corresponds to value of parameter 𝜍𝑖 equal to unity. The

maximum values of the other curves (Figure 2.3, left) on either side of it, decrease as the

distance.

As it is shown in Figure 2.3 (left), there are a pair of curves, maximum values of

which are located on a same level. These curves correspond to a pair of mutually 𝜍𝑖

parameter reverse values. Thus, as it follows from Figure 2.3 and Table 1: 𝜍1 ≈ 1 𝜍16⁄ and

𝜍2 ≈ 1 𝜍8⁄ . Similar phenomena occur also in a two-layer cylindrical waveguide. The

difference is in the value of the parameter 𝜍𝑖: 𝜍 =1

3𝑍0𝜎1𝑑 for planar and 𝜍 =

1

√3𝑍0𝜎1𝑑 for

cylindrical cases, i.e. √3 times less. Recall that the resonance frequencies in the planar

case to √2 times less than in the corresponding cylindrical: 𝜔𝑟𝑒𝑧 = 𝑐√1 𝑎1𝑑⁄ (𝑎1 inner half

- 41 -

height of planar structure) instead of 𝜔𝑟𝑒𝑧 = 𝑐√2 𝑎1𝑑⁄ (𝑎1 inner radius of cylinder). Such

relations for broadband resonance frequencies have been identified in [65] for the single-

layer resistive two-dimensional flat and cylindrical structures (Figure 2.4). For comparison,

Figure 2.4 shows the longitudinal impedance distributions for a two-layer metal tube with

an ideally conducting outer layer. Its geometric and electromagnetic parameters are

brought into correspondence with the parameters of the flat two-layer metal structure

considered above. The internal radius of the cylinder 𝑟0 = 1 𝑐𝑚, the conductivity of the

inner layer is 𝜎 = 2 × 103 Ω−1𝑚−1, and its thickness varies between 1 𝜇𝑚 and 5 𝜇𝑚.

Figure 2.4: Real (left) and imaginary (right) parts of the longitudinal impedance of two-layer

cylindrical structure. On the graphs, the thicknesses of the inner layer in microns are shown.

The wake potential is defined as the inverse Fourier transform of impedance (2.43)

𝑊∥(𝑥, 𝑦, 𝑠) = ∫ 𝑍∥(𝑥, 𝑦, 𝑘)∞

−∞𝑒−𝑗𝑘𝑠𝑑𝑠, 𝑠 = 𝑧 − 𝑣𝑡 (2.47)

Figure 2.5 shows the distribution of the longitudinal wake potential for three different

thicknesses of the inner layer at the fixed conductivity. In all three cases the wake

potentials are quasi-periodic functions with the period Δ𝑠 given by the resonant frequency

Δ𝑠 = 𝑐 𝑓𝑟𝑒𝑧⁄ , 𝑓𝑟𝑒𝑧 =𝑐

2𝜋√1

𝑎𝑑. The latter indicates the particle radiation monochromaticity

similar to two-layer cylindrical metal waveguide [50].

- 42 -

Figure 2.5: Longitudinal wake potential of the two-layer flat structure for different values of

the thickness of the inner layer and the fixed conductivity 𝜎 = 2 × 103 Ω−1𝑚−1; half-height of

structure 𝑎 = 1 𝑐𝑚. For comparison, the wake function for two unbounded infinite single-layer

plates with halt-width 𝑎 = 1 𝑐𝑚 and 𝜎 = 2 × 103 Ω−1𝑚−1 is given (dashed curve).

Figure 2.6 shows the wake potential distributions for a two-layer cylindrical structure.

Figure 2.6: Longitudinal wake potential of the two-layer cylindrical waveguide for different

values of the thickness of the inner layer and the fixed conductivity 𝜎 = 2 × 103 Ω−1𝑚−1;

inner radius of cylinder is equal to half height of flat structure (Figure 2.5) 𝑟0 = 𝑎 = 1 𝑐𝑚. The

designations of the curves correspond to the thicknesses of the inner coating in microns. The

wake potential for unbounded single-layer resistive cylinder with radius 𝑟0 = 1 𝑐𝑚 and 𝜎 =

2 × 103 Ω−1𝑚−1 is given in addition (dashed curve).

- 43 -

As in the plane case, the wake function of a cylindrical structure is a quasi-

monochromatic oscillating damped function with a single oscillation frequency 𝑓𝑟𝑒𝑧 =𝑐

2𝜋√2

𝑎𝑑.

The period of the oscillations is Δ𝑠 = 𝑐 𝑓𝑟𝑒𝑧⁄ (√2 times less of one of flat case).

Comparing the curves for impedances and wake potentials for a two-layered

cylindrical structure, we notice from Figure 2.4 (curve with label “2.5”), that in the

frequency representation the maximum quality factor (maximum amplitude) of the

impedance reaches at the thickness of the inner coating equal to 2.5 𝜇𝑚. In the space-time

representation this correlates with the minimum attenuation of the wake (Figure 2.6, curve

with label “2.5”). In the cylindrical case, the attenuation coefficient is proportional to the

value of 휁 = 𝜉 + 𝜉−1, with 𝜉 =1

√3𝑍0𝑑𝜎 and attains its minimum value at 𝜉 = 1 [50].

The above patterns (Figure 2.3 and Figure 2.5) are revealed in the case of fixed

value of the inner layer conductivity. The regularities of the structure with a fixed thickness

of the inner layer, are denoted in Figure 2.7. The Figure shows the distribution of the

amplitude values of the impedance curve at 𝑑 = 1𝜇𝑚 and 1500 Ω−1𝑚−1 < 𝜎 <

11000 Ω−1𝑚−1, which corresponds to the range of variation of the parameter 𝜉 =1

√3𝑑𝜎𝑍0

within 0.3 < 𝜉 < 2.5.

- 44 -

Figure 2.7: Distribution of resonance values of the two-layer flat structure impedance as a function

of the conductivity 𝜎 of the inner layer for fixed geometric parameters of the structure (𝑎 =

1𝑐𝑚, 𝑑 = 1 𝜇𝑚). The horizontal dashed lines indicate the mutually conjugate values of the

conductivities 𝜎𝑎 and 𝜎𝑏 for which 휁𝑏 = 1 휁𝑎⁄ with 휁𝑎(𝑏) = 3−1 2⁄ 𝑑𝑍0𝜎𝑎(𝑏). Vertical dashed line

indicates the maximum possible level of impedance for a given thickness of the inner layer 𝑑 =

1𝜇𝑚.

A significant difference between the cylindrical and flat cases is that in the first case

the optimal condition (in the sense of the maximum Q of the system) is the fulfillment of

equality 𝜉 =1

√3𝑍0𝜎𝑑 = 1. In the second case, when fixing the conductivity value of the

inner layer σ, the condition for optimizing the structure is the equality 𝜉 =1

3𝑍0𝜎𝑑 = 1, i.e.

the optimum layer thickness in this case is determined through its conductivity 𝜎 by the

expression 𝑑 =3

𝑍0𝜎. In other hand, fixation of the thickness of the inner layer 𝑑 leads to the

optimization condition 𝜉 =1

√3𝑍0𝜎𝑑 = 1 and to the optimal conductivity of the layer 𝜎 =

√3

𝑍0𝑑.

Thus, for a given conductivity of the inner layer, in order to select the optimum thickness of

the layer, it is necessary to use the relation 𝑑 =3

𝑍0𝜎, and for a given layer thickness d, its

optimal conductivity is determined by the expression 𝜎 =√3

𝑍0𝑑.

- 45 -

Figure 2.8: Optimization of the parameters of a two-dimensional two-layer metal structure with the

initial fixation of the thickness 𝑑 = 1𝜇𝑚 of the inner layer.

Figure 2.8 shows the results of the sequential optimization process with the initial

fixation value of the thickness of the inner layer at the level of 𝑑 = 1𝜇𝑚. The figure shows

5 curves depicting the distribution of longitudinal impedances for the half-height of

structure 𝑎 = 1𝑐𝑚. Curve 1 (black solid curve) is plotted for the thickness of the inner layer

𝑑 = 1 𝜇𝑚 with a conductivity 𝜎 = 4594 Ω−1𝑚−1 , determined by the formula 𝜎 =√3

𝑍0𝑑, i.e.

curve 1 represents the impedance with the highest possible amplitude level, achievable at

𝑑 = 1 𝜇𝑚 by means of the corresponding selection of the conductivity of the inner layer.

Fixing, then, the received value of conductivity, by the formula 𝑑2 =3

𝑍0𝜎 we determine for it

the optimum value of layer thickness 𝑑2 = 1.732 𝜇𝑚 , wich is optimal for this value of

conductivity. As a result, we obtain curve 2 (red solid curve). Optimization leads to an

increase in the thickness of the inner layer and, consequently, to a decrease in the

resonant frequency. Further optimization again assumes the use of the formula 𝜎2 =√3

𝑍0𝑑.

As a result we have curve 3 with reduced conductivity 𝜎3 = 2653 Ω−1𝑚−1 and a resonance

conserved frequency. This procedure, consisting in the successive application of the

formulas 𝑑𝑖 =3

𝑍0𝜎𝑖 and 𝜎𝑖+1 =

√3

𝑍0𝑑𝑖, can be repeated many times, successively increasing

the thickness of the inner layer and its conductivity and decreasing the resonance

frequency (Figure 2.8).

- 46 -

An alternative way to perform a series of optimizations is shown in Figure 2.9. It is

obtained by first fixing the conductivity of the inner layer of the structure. In the figure,

curve “1” corresponds to the originally selected conductivity of the inner layer 𝜎 = 2 ×

103 Ω−1𝑚−1. Its thickness is optimized using the formula 𝑑 =3

𝑍0𝜎= 3.98 𝜇𝑚. Curve “2” (red,

dashed) is obtained by optimizing the conductivity for the resulting length: 𝜎2 =√3

𝑍0𝑑=

1154 Ω−1𝑚−1 . The resonant frequency does not change in this case. The further

optimization step, corresponding to curve “3”, is carried out using the formula 𝑑2 =3

𝑍0𝜎2=

6.89 𝜇𝑚. This optimization stage is accompanied by an increase in the thickness of the

inner layer and by a decrease of resonant frequency. At the next stage, the optimum

conductivity value is determined for the layer new thickness 𝜎3 =√3

𝑍0𝑑2= 385 Ω−1𝑚−1 (curve

“3”) and so on.

Figure 2.9: Optimization of the parameters of a two-dimensional two-layer metal structure with the

initial fixation of the inner layer conductivity 𝜎 = 2 × 103 Ω−1𝑚−1.

Thus, there are two different possibilities to optimize a two-dimensional two-layer

metal structure: the first (Figure 2.8) is accompanied by an increase of the conductivity of

the material of the inner layer, while the second (Figure 2.9) lowers it.

The practical significance of the above study is the ability to correctly select the

thickness of the inner layer and the electromagnetic characteristics of the material filling it.

- 47 -

This, for example, was successfully done during of experimentally testing the resonant

properties of a two-layer metal structure [67] with severe limitations on the measuring

equipment, whose frequency range was limited by 14 GHz bandwidth.

Figure: 2.10: Longitudinal impedance of cylindrical pipe with optimized parameters: optimal inner

layer width 𝑑 = 2.3𝜇𝑚 for given conductivity 𝜎 = 2000 Ω−1𝑚−1 (red); optimal inner layer

conductivity 𝜎 = 4600 Ω−1𝑚−1 for given layer width 𝑑 = 1𝜇𝑚 (black).

In the case of a two-layered cylindrical structure, the picture is much simpler. In this

case, the narrow-band resonance character of the impedance curve is completely due to

the proximity to the unity of the parameter 𝜉 =1

√3𝑍0𝜎𝑑 [50]. The impedance curve is

completely determined by setting two parameters (𝜉, 𝜎; 𝜉, 𝑑 or 𝜎, 𝑑) of three (𝜉, 𝜎, 𝑑).

2.5 Approximate analytical representation of impedance

The integral representation of the longitudinal impedance (Eq. 2.43 – Eq. 2.45) is

accurate (exact) and, as was shown above, it is possible in principle to calculate both the

impedance and the wake potential (represented in the form of a double integral (Eq. 2.47)).

For calculation, a corresponding software package has been developed by using the

software package Mathematica8. The integral form of the expressions for the impedance

and the wake function, as well as the complexity of the integrand expression, leads,

however, to some difficulties in interpreting them. Integration by means of deformation of

the contour of integration and its transfer to the complex plane is associated with taking

- 48 -

into account, in addition to complex poles of denominator, also integration along the cuts

on the Riemann surface. The latter circumstance is connected with the non-analyticity of

the function 𝑘𝑦(1) = √𝑘𝑥2 + (1 − 휀

(1)𝜇(1)𝑐2)𝑘2, which leads to the non-analyticity of the

denominator of the integrand as a whole.

The analytizing of the integrand in (Eq. 2.43) is performed taking into account the

smallness of the parameter 𝑑 (𝑑 ≪ 𝑎), characterized the thickness of the inner layer. For

this purpose, we rewrite the function 𝐴1𝐶 (Eq. 2.44) in the form

𝐴1𝐶 =𝑞𝑍0𝑘𝑥

8𝜋2𝑍0𝜎𝑘

�̃�, �̃� = �̃�1

𝐶�̃�2𝐶 (2.48)

where now

�̃�1𝐶 = 𝑘𝑦

(1)𝐶ℎ(𝑘𝑥𝑎)𝐶𝑡ℎ (𝑘𝑦

(1)𝑑) + 𝑘𝑥𝑆ℎ(𝑘𝑥𝑎)

�̃�2𝐶 = 𝑘𝑥(𝑘 + 𝑗 𝑍0𝜎)𝐶ℎ(𝑘𝑥𝑎) + 𝑘𝑘𝑦

(1)𝑆ℎ(𝑘𝑥𝑎)𝑇ℎ (𝑘𝑦

(1)𝑑)

(2.49)

After multiplication, the denominator in Eq. 2.48 takes the form

�̃� = 𝑘𝑦(1)𝑘𝑥(𝑘+ 𝑗 𝑍0𝜎)𝐶ℎ

2(𝑘𝑥𝑎)𝐶𝑡ℎ (𝑘𝑦

(1)𝑑)+

1

2(𝑘𝑦

(1)2𝑘+ 𝑘𝑥

2(𝑘+ 𝑗 𝑍0𝜎))𝑆ℎ(2𝑘𝑥𝑎)+𝑘𝑦

(1)𝑘𝑥𝑘𝑆ℎ

2(𝑘𝑥𝑎)𝑇ℎ (𝑘𝑦

(1)𝑑)

(2.50)

Further, we expand �̃� in a series with respect to the small parameter 𝑑 and leave

the first three terms of the expansion (𝑑−1, 𝑑0 and 𝑑). As a result, we get

�̃� ≈ 𝑢−1 𝑑⁄ + 𝑢0 + 𝑢1𝑑 (2.51)

where

𝑢1 =1

3𝑘𝑦(1)2𝑘𝑥 ((𝑘 + 𝑗 𝑍0𝜎)𝐶ℎ

2(𝑘𝑥𝑎) + 3𝑘𝑆ℎ2(𝑘𝑥𝑎))

𝑢0 =1

2(𝑘𝑦

(1)2𝑘 + 𝑘𝑥

2(𝑘 + 𝑗 𝑍0𝜎))𝑆ℎ(2𝑘𝑥𝑎)

𝑢−1 = 𝑘𝑥(𝑘 + 𝑗 𝑍0𝜎)𝐶ℎ2(𝑘𝑥𝑎)

(2.52)

Expression 2.51 with taking into account Eq. 2.52 no longer contains square roots

of the integration variables and is analytical.

- 49 -

Believing, further 𝑘𝑥2 ≪ 𝑍0𝜎𝑘, (in the cylindrical case this approximation is equivalent

to approximation for the permittivity of the metal 휀 ≈ 𝑗 𝜎 𝜔⁄ , which is valid for not very large

frequencies) we have:

𝑘𝑦(1)2

≈ −𝑗𝑍0𝜎𝑘 (2.53)

and instead of Eq. 2.52, we can write

𝑢1 = −1

3𝑗𝑍0𝜎𝑘𝑘𝑥((𝑘 + 𝑗 𝑍0𝜎)𝐶ℎ

2(𝑘𝑥𝑎) + 3𝑘𝑆ℎ2(𝑘𝑥𝑎))

𝑢0 =1

2(𝑘𝑘𝑥

2 + 𝑗 𝑍0𝜎(𝑘𝑥2 − 𝑘2))𝑆ℎ(2𝑘𝑥𝑎)

𝑢−1 = 𝑘𝑥(𝑘 + 𝑗 𝑍0𝜎)𝐶ℎ2(𝑘𝑥𝑎)

(2.54)

Figures (2.11–2.15) show comparison of calculations for exact formulae and in

approximation (Eq. 2.51 – Eq. 2.54) for three different half-widths of a plane structure: 𝑎 =

10 𝑐𝑚 (Figure 2.11 and Figure 2.12), 𝑎 = 1𝑐𝑚 (Figure 2.13 and Figure 2.14) and 𝑎 = 1 𝑚𝑚

(Figure 2.15). The thickness of the inner layer is chosen equal to 7 𝜇𝑚. Dashed red curves

represent the results obtained from exact formulae. The approximate results are

represented by solid black curves.

Fig. 2.11. Real (left) and imaginary (right) components of the longitudinal impedance of a

two-layer metal structure, calculated by exact formulae (red, dashed) and using the approximation

(2.48, 2.51) (black, solid): 1 - 𝜎 = 2 × 104 Ω−1𝑚−1, 휁 = 30.5; 2 - 𝜎 = 2 × 103 Ω−1𝑚−1, 휁 = 3.05; 3 -

𝜎 = 656 Ω−1𝑚−1, 휁 = 1; 𝑎 = 10 𝑐𝑚, 𝑑 = 7 𝜇𝑚.

- 50 -

Fig. 2.13. Real (left) and imaginary (right) components of the longitudinal impedance of a two-layer

metal structure, calculated by exact formulae (red, dashed) and using the approximation (2.48,

2.51) (black, solid): 1 - 𝜎 = 2 × 104 Ω−1𝑚−1, 휁 = 30.5; 2 - 𝜎 = 2 × 103 Ω−1𝑚−1, 휁 = 3.05; 3 - 𝜎 =

656 Ω−1𝑚−1, 휁 = 1; 𝑎 = 1 𝑐𝑚, 𝑑 = 7 𝜇𝑚.

As can be seen from Figure 2.11 and Figure 2.13, for relatively large values of the

half-widths of the structure (1 𝑐𝑚 ≤ 𝑎 ≤ 10 𝑐𝑚) there is a complete coincidence of the

approximation calculations with the exact ones even for much larger (in comparison with

unity) values of the parameter 휁 =1

√3𝑍0𝑑 (up to 휁~30). Significant discrepancies occurs at

large values of this parameter 휁~100 (Figure 2.12 and Figure 2.14).

Fig. 2.12. Real (left) and imaginary (right) components of the longitudinal impedance of a two-layer

metal structure, calculated by exact formulae (red, dashed) and using the approximation (2.48,

2.51) (black, solid): 1 - 𝜎 = 2 × 105 Ω−1𝑚−1, 휁 = 135 ; 2 - 𝜎 = 2 × 106 Ω−1𝑚−1, 휁 = 1523 ; 𝑎 =

10 𝑐𝑚, 𝑑 = 7 𝜇𝑚.

- 51 -

Fig. 2.14. Real (left) and imaginary (right) components of the longitudinal impedance of a two-layer

metal structure, calculated by exact formulae (red, dashed) and using the approximation (2.48,

2.51) (black, solid): 1 - 𝜎 = 2 × 105 Ω−1𝑚−1, 휁 = 135 ; 2 - 𝜎 = 2 × 106 Ω−1𝑚−1, 휁 = 1523 ; 𝑎 =

1 𝑐𝑚, 𝑑 = 7 𝜇𝑚.

As follows from the Figure 2.15, a decrease in the gap between the plates leads to

a toughening of the requirement of 휁 be close to unity. Thus, for 휁 = 30 (curve “1” on

Figure 2.15) already there is an essential discrepancy between the exact and approximate

solution, whereas for 휁 = 7.6 (curve “2” on Figure 2.15) they practically coincide.

Fig. 2.15. Real (left) and imaginary (right) components of the longitudinal impedance of a two-layer

metal structure, calculated by exact formulae (red, dashed) and using the approximation (2.48,

2.51) (black, solid): 1 - 𝜎 = 2 × 104 Ω−1𝑚−1, 휁 = 30.5; 2 - 𝜎 = 5 × 103 Ω−1𝑚−1, 휁 = 7.6. 𝑎 = 1 𝑚𝑚,

𝑑 = 7 𝜇𝑚.

Further simplification of the integrand (Eq. 2.50 – Eq. 2.53) is achieved by replacing

the hypergeometric functions by their decomposition. The expansions in a series of

hypergeometric functions 𝑆ℎ(𝑥) and 𝐶ℎ(𝑥) have the form [68]

- 52 -

𝑆ℎ(𝑥) = ∑𝑥2𝑛−1

(2𝑛−1)!∞𝑛=1 , 𝐶ℎ(𝑥) = ∑

(𝑥)2𝑛−2

(2𝑛)!∞𝑛=1 (2.55)

Substituting Eq. 2.55 in Eq. 2.53 and leaving the first N terms of the expansion in

them, we have instead of Eq. 2.48

𝐴1𝐶 =𝑞𝑍0

8𝜋2𝑍0𝜎𝑘

�̃�′ (2.56)

where

�̃�′ ≈ 𝑢−1′ 𝑑⁄ + 𝑢0

′ + 𝑢1′𝑑 (2.57)

and

𝑢1′ =

1

6𝑗𝑘𝜎𝑍0((2𝑘 − 𝑖𝜎𝑍0) − (4𝑘 + 𝑖𝜎𝑍0)𝐶ℎ𝑁(2𝑎𝑘𝑥))

𝑢−1′ =

1

2(𝑘 + 𝑗 𝑍0𝜎)(1 + 𝐶ℎ𝑁(2𝑎𝑘𝑥)

𝑢0′ =

1

2(𝑘𝑘𝑥

2 + 𝑗 𝑍0𝜎(𝑘𝑥2 − 𝑘2))

𝑆ℎ𝑁(2𝑘𝑥𝑎)

𝑘𝑥

(2.58)

with

𝑆ℎ𝑁(𝑥) = ∑𝑥2𝑛−1

(2𝑛−1)!𝑁𝑛=1 , 𝐶ℎ𝑁(𝑥) = ∑

𝑥2𝑛−2

(2𝑛)!𝑁𝑛=1 (2.59)

In this approximation, the denominator (Eq. 2.57) is represented in the form of an

algebraic polynomial of 2N-th order in powers of 𝑘𝑥

�̃�′ = ∑ 𝑎𝑛𝑘𝑥𝑛2𝑁

𝑛=0 (2.60)

and can be written in a form containing its complex roots 𝑥𝑖 , 𝑖 = 1,2, … 2𝑁

�̃�′ = 𝑎2𝑁∏ (𝑘𝑥 − 𝑥𝑖)2𝑁𝑖=1 (2.61)

Thus, Eq. 2.56 can be rewritten as follows

𝐴𝐶 =𝑞𝑍0

8𝜋2𝑍0𝜎𝑘

𝑎2𝑁∏ (𝑘𝑥−𝑥𝑖)2𝑁𝑖=1

(2.62)

The longitudinal impedance, which is an integral of this expression over 𝑘𝑥, can be

expressed explicitly in terms of the roots 𝑥𝑖 of the equation �̃�′(𝑘𝑥) = 0

- 53 -

𝑍|| = ∫ 𝐴𝐶𝑑𝑘𝑥∞

−∞= −

𝑍02𝜎𝑘

4𝜋2𝑎2𝑁∑

𝑙𝑛{−𝑥𝑛}

∏ (𝑥𝑛−𝑥𝑖)2𝑁𝑖=1𝑖≠𝑛

2𝑁𝑛=1 (2.63)

The roots 𝑥𝑖 of the equation �̃�′(𝑘𝑥) = 0 are determined numerically for an arbitrary

value of N with the help of the software package Mathematica.

Longitudinal impedance of a two-layer metal structure calculated by exact and

approximate formulas formulae is presented in Figure 2.16. In the figure black solid

impedance curves are calculated using the approximate equations 2.51 and 2.53, blue

dashed ones are from Eq. 2.63 for 𝑛 = 5 and red dashed ones are for calculated by exact

formula.

Fig. 2.16. Real (left) and imaginary (right) components of the longitudinal impedance of a two-layer

metal structure, calculated by exact formulae (red, dashed), using the approximation (2.48, 2.51)

(black, solid) and using approximation (2.60) for n=5 (blue, dashed): 1 - 𝜎 = 2 × 104 Ω−1𝑚−1, 휁 =

30.5; 2 - 𝜎 = 5 × 103 Ω−1𝑚−1, 휁 = 7.6; 𝑎 = 1 𝑚𝑚, 𝑑 = 7 𝜇𝑚.

In Figure 2.16, the impedance curves calculated using the asymptotic (Eq. 2.62) are

superimposed on the impedance curves shown in Fig. 2.15.

2.6 Summary

Method of calculating the longitudinal impedance of a multilayer two-dimensional

flat structure is presented. The matrix formalism that allows to couple electromagnetic field

components in the inner and outer regions of the structure has been developed. Non-

ultrarelativistic particle radiation fields in the multilayer flat symmetric and asymmetric

structure are obtained. Explicit expressions of the radiation fields in two-layer structure

- 54 -

with unbounded external walls and in single-layer unbounded structure are derived as a

special case of fields in multilayer structure.

An ultrarelativistic particle radiation in a symmetrical two-layer flat structure with

perfectly conducting outer layer is derived. Longitudinal impedance of this structure with

low conductivity thin inner layer and high conductivity thick outer layer wall is obtained

numerically. It is shown that the impedance has a narrow-band resonance. The resonance

frequency dependence of a structure parameters is obtained. Parameter 𝜍𝑖 =1

3𝑍0𝜎1𝑑𝑖 is

introduced, which describes maximum values of impedance. It is shown that impedances

of structures with reverse 𝜍 parameters (𝜍𝑖 ∗ 𝜍𝑗 = 1) has the same maximum values and it

reaches the highest maximum when 𝜍𝑗 is equal to unity.

- 55 -

Chapter 3: Rectangular resonator with two-layer vertical walls

3.1 Introduction

In this chapter, dispersion relations are obtained for the electromagnetic eigen-

oscillations in a rectangular resonator with ideally conducting walls. The horizontal (top

and bottom) walls of this resonator are covered from the inside by a thin low-conductive

layer.

On the second Chapter it is shown that double-layered parallel plates for appropriate

parameters have a distinct, narrow band resonance. In this Chapter rectangular copper

cavity with double-layered horizontal walls is observed. Here, unlike of the parallel plates,

one has a discrete set of wavenumbers in all three degree of freedom. Compared with

two-dimensional structure, the pattern of modal frequency distribution in such a resonator

is distorted: in addition to the fundamental resonance frequency, a set of secondary

resonances caused by side walls exists. It is important to note the presence of the

fundamental resonance, fixed by measurements, and caused by the two-layer horizontal

cavity walls. To be able to distinguish the main mode from secondary resonances, finite

wall resonator model is developed and resonance frequencies of the resonator are

calculated.

In this Chapter secondary resonances caused by side walls are obtained. Criterions

of resonator materials choice are interpreted and resonance frequencies are predicted via

the resonator model with ideal conducting walls. The choice is conditioned by the

frequency bandwidth of the experimental set-up ( 14 𝐺𝐻𝑧 ) and the predicted single

resonance for two parallel laminated plates (chapter 2). The electro-dynamical properties

of test resonator model (perfectly conducting rectangular cavity with inner low conducting

layers) are studied. Resonance frequencies of the test structure are measured

experimentally and comparison with predicted resonant frequencies of resonator model

and two parallel double-layer plates is done. Good agreement between these results is

obtained.

- 56 -

Experimentally obtained resonance frequencies are subjected to the laws specific to

this kind of structures: their value decreases with increasing height of the cavity. At the

same time, the difference between the measured and calculated values increases.

Decreasing a structure height, the contribution of its sidewalls also decrease, that’s why for

a structures with small height there is a good agreement between experimental and

theoretical resonant frequencies.

3.2 IHLCC cavity with ideally conductive walls

To obtain dispersion relations analytically for a tested cavity, a simplified resonator

model is studied. It is assumed that all four sidewalls of IHLCC cavity are ideally

conductive materials and upper and lower ideally conductive walls are covered with a thin

layer of low-conductive material (Fig 3.1).

Figure 3.1: IHLCC cavity shape.

This model is very close to the cavity with copper walls, so it is expected that

analytical results would be in a good agreement with tested cavity results. Exact dispersion

equations may be obtained for a similar structure by the method of partial areas. In this

case, three regions may be distinguished:

1. 0 ≤ 𝑥 ≤ 𝑏, 0 ≤ 𝑧 ≤ 𝑙 top low conductivity material plate

(3.1) 2. 0 ≤ 𝑥 ≤ 𝑏, 0 ≤ 𝑧 ≤ 𝑙 vacuum chamber of cavity

3. 0 ≤ 𝑥 ≤ 𝑏, 0 ≤ 𝑧 ≤ 𝑙 lower low conductivity material plate

- 57 -

Electric and magnetic fields of cavity in each of the three regions are sought in the

following form

�⃗⃗�𝑖 = �⃗⃗�𝑖(1)+ �⃗⃗�𝑖

(2), �⃗⃗�𝑖 = (𝑗𝑘𝑐)

−1𝑟𝑜𝑡�⃗⃗�𝑖, 𝑖 = 1,2,3 (3.2)

with

𝐸𝑥,𝑖(1)

𝐸𝑥,𝑖(2)} = {

𝐴𝑥,𝑖(1)𝐶ℎ(𝑄𝑖𝑦)

𝐴𝑥,𝑖(2)𝑆ℎ(𝑄𝑖𝑦)

} 𝐶𝑜𝑠(𝑘𝑥𝑥)𝑆𝑖𝑛(𝑘𝑧𝑧)

𝐸𝑦,𝑖(1)

𝐸𝑦,𝑖(2)} = {

𝐴𝑦,𝑖(1)𝑆ℎ(𝑄𝑖𝑦)

𝐴𝑦,𝑖(2)𝐶ℎ(𝑄𝑖𝑦)

} 𝑆𝑖𝑛(𝑘𝑥𝑥)𝑆𝑖𝑛(𝑘𝑧𝑧)

𝐸𝑧,𝑖(1)

𝐸𝑧,𝑖(2)} = {

𝐴𝑧,𝑖(1)𝐶ℎ(𝑄𝑖𝑦)

𝐴𝑧,𝑖(2)𝑆ℎ(𝑄𝑖𝑦)

} 𝑆𝑖𝑛(𝑘𝑥𝑥)𝐶𝑜𝑠(𝑘𝑧𝑧)

(3.3)

where 𝐴𝑥,𝑦,𝑧,𝑖(1,2)

are arbitrary amplitudes, 𝑄1,3 = 𝐾,𝑄2 = 𝑘𝑦 and

𝐾 = √𝑘2(1 − 휀′𝜇′) + 𝑘𝑦2, 𝑘 =

𝜔

𝑐= √𝑘𝑥

2 + 𝑘𝑧2 − 𝑘𝑦

2, (3.4)

The boundary conditions on the perfectly conducting sidewalls are

𝐸𝑥,𝑖 = 𝐸𝑦,𝑖 = 0

𝐸𝑦,𝑖 = 𝐸𝑧,𝑖 = 0 at

𝑧 = 0, 𝑧 = 𝑙𝑥 = 0, 𝑥 = 𝑏

, 𝑖 = 1,2,3 (3.5)

To satisfy these conditions horizontal wavenumbers should be in the form of 𝑘𝑥 =

(𝑚 − 1)𝜋

𝑏, 𝑘𝑧 = (𝑛 − 1)

𝜋

𝑙, 𝑚, 𝑛 = 1,2,3…. 𝑘𝑦 transverse vertical wavenumber should be

determined from boundary condition on low-conductive layer surfaces and vertical walls.

On vacuum-low conductive layer borders, from the continuity of tangential electric

and magnetic field components

𝐸𝑥,1 = 𝐸𝑥,2, 𝐸𝑧,1 = 𝐸𝑧,2𝐵𝑥,1 = 𝐵𝑥,2, 𝐵𝑧,1 = 𝐵𝑧,2

at 𝑦 = 𝑎/2 (3.6)

𝐸𝑥,2 = 𝐸𝑥,3, 𝐸𝑧,2 = 𝐸𝑧,3𝐵𝑥,2 = 𝐵𝑥,3, 𝐵𝑧,2 = 𝐵𝑧,3

at 𝑦 = −𝑎/2 (3.7)

- 58 -

On the outer boundaries between low conductive layers and ideal conductors,

electrical tangential components of fields must vanish

𝐸𝑥,1 = 0, 𝐸𝑧,1 = 0

𝐸𝑥,3 = 0, 𝐸𝑧,3 = 0 at

𝑦 = 𝑎′/2

𝑦 = −𝑎′/2 (3.8)

Using Maxwell's equation 𝑑𝑖𝑣�⃗⃗�𝑖(1,2) = 0, the amplitudes 𝐴𝑧,𝑖

(1,2) can be expressed by

the amplitudes 𝐴𝑥,𝑖(1,2)

and 𝐴𝑦,𝑖(1,2)

. Thus, we have a linear homogeneous system consisting

of 12 equations (Eq. 3.6 - Eq. 3.8) and containing the same number of indeterminate

coefficients: 𝐴𝑥,𝑖(1,2)

, 𝐴𝑦,𝑖(1,2)

with 𝑖 = 1,2,3. Condition of existence of non-trivial solutions of this

system is the vanishing of its determinant

𝐷 = 𝑃1𝑃2𝑃3𝑃4 = 0 (3.9)

where

𝑃1 = 𝐾𝑛𝑚𝐶ℎ(𝑑𝐾)𝑆ℎ(𝑎𝑘𝑦𝑓) + 𝑘𝑦𝑓𝑆ℎ(𝑑𝐾𝑛𝑚)𝐶ℎ(𝑎𝑘𝑦𝑓) (3.10a)

𝑃2 = 𝐾𝑛𝑚𝐶ℎ(𝑑𝐾𝑛𝑚)𝐶ℎ(𝑎1𝑘𝑦𝑓) + 𝑘𝑦𝑓𝑆ℎ(𝑑𝐾𝑛𝑚)𝑆ℎ(𝑎1𝑘𝑦𝑓) (3.10b)

𝑃3 = −𝐾𝑛𝑚𝑘2𝑆ℎ(𝑑𝐾𝑛𝑚)𝐶ℎ(𝑎𝑘𝑦𝑓) − 𝑘𝑦𝑓𝑘

2휀′𝜇′𝐶ℎ(𝑑𝐾𝑛𝑚)𝑆ℎ(𝑎𝑘𝑦𝑓) (3.10c)

𝑃4 = −𝑘𝑦𝑓𝑘2휀′𝜇′𝐶ℎ(𝑑𝐾𝑛𝑚)𝐶ℎ(𝑎𝑘𝑦𝑓) −𝐾𝑛𝑚𝑘

2𝑆ℎ(𝑑𝐾𝑛𝑚)𝑆ℎ(𝑎𝑘𝑦𝑓) (3.10d)

where 𝐾𝑛𝑚 = √𝑘𝑥𝑛2 + 𝑘𝑧𝑚

2 − 𝑘2휀′𝜇′, 𝑘 =𝜔

𝑐= √𝑘𝑥𝑛

2 + 𝑘𝑧𝑚2 − 𝑘𝑦𝑓

2 , 𝑛,𝑚, 𝑓 =

1,2,3,….

The result is four independent dispersion equations:

𝑃𝑖 = 0, 𝑖 = 1,2,3,4 (3.11)

Each of these four equations includes three eigenvalues: 𝑘𝑥𝑛, 𝑘𝑧𝑚 and 𝑘𝑦𝑓 .

Moreover, the horizontal transverse 𝑘𝑥𝑛 and longitudinal 𝑘𝑧𝑚 wave numbers are fixed for

given numbers 𝑛 and 𝑚. They form two infinite sequences generated by integer indices 𝑚

and 𝑛 . Thus, each of equations (Eq. 3.10.a -3.10d) defines the sequence of 𝑘𝑦𝑓 , (𝑓 =

1,2,3, … ) values for each fixed combination of indices 𝑚 and 𝑛. As a result, from these

- 59 -

equations we get dependent on 𝑚 and 𝑛 sequences of eigenvalues 𝑘𝑦 = 𝑘𝑚𝑛𝑓(𝑖) (𝑖 = 1,2,3,4)

for = 1,2,3… .

Let us compare the obtained expressions with the corresponding expressions,

obtained in Chapter 1, for the two-layer flat structure (1.24):

𝑇1 = 𝑘𝑦(1)𝐶ℎ(𝑑𝑘𝑦

(1))𝑆ℎ(𝑎𝑘𝑦(0)) + 𝑘𝑦

(0)𝑆ℎ(𝑑𝑘𝑦(1))𝐶ℎ(𝑎𝑘𝑦

(0))

𝑇2 = 𝑘𝑦(1)𝐶ℎ(𝑑𝑘𝑦

(1))𝐶ℎ(𝑎𝑘𝑦(0)) + 𝑘𝑦

(0)𝑆ℎ(𝑑𝑘𝑦(1))𝑆ℎ(𝑎𝑘𝑦

(0))

𝑇3 = −𝑘𝑦(0)𝑘2휀𝑖

′𝜇𝑖′𝐶ℎ(𝑑𝑘𝑦

(1))𝑆ℎ(𝑎𝑘𝑦(0)) − 𝑘𝑦

(1)𝑘2𝑆ℎ(𝑑𝑘𝑦(1))𝐶ℎ(𝑎𝑘𝑦

(0))

𝑇4 = −𝑘𝑦(0)𝑘2휀𝑖

′𝜇𝑖′𝐶ℎ(𝑑𝑘𝑦

(1))𝐶ℎ(𝑎𝑘𝑦(0)) − 𝑘𝑦

(1)𝑘2𝑆ℎ(𝑑𝑘𝑦(1))𝑆ℎ(𝑎𝑘𝑦

(0))

(3.12)

Obviously, there is a correspondence: 𝑃𝑖 ↔ 𝑇𝑖 , 𝑖 = 1,2,3,4 . A complete coincidence

cannot take place, since in the case of an infinite two-dimensional structure, the

variables 𝑘𝑥 and 𝑘𝑧 are integration variables, but in the case of a resonator, the

wave numbers 𝑘𝑥𝑛 and 𝑘𝑧𝑚 are discrete and fixed for specified values by the index

n and m.

The standing wave excited in the resonator consists of the imposition of two

traveling waves propagating in opposite directions: a wave incident on the front wall of the

resonator (�⃗⃗�1) and reflected from it (�⃗⃗�2):

𝐸𝑥,𝑦,𝑧 = 𝐸1,𝑥,𝑦,𝑧 ± 𝐸2,𝑥,𝑦,𝑧, 𝐸1,2,𝑥,𝑦,𝑧 = 𝐴1,2,𝑥,𝑦,𝑧𝑒𝑗(±𝑘𝑧𝑧−𝜔𝑡) (3.13)

The wave �⃗⃗�1 can serve as an analog of a wave propagating in an infinite planar two-layer

structure, generated by a particle. Since its frequency 𝜔 in this case is complex, it can be

represented in the form:

𝜔 = 𝑐√𝐴 + 𝑗𝐵 = 𝑐√√𝐴2+𝐵2+𝐵

2+ 𝑗𝑐 𝑆𝑖𝑔𝑛 𝐵√

√𝐴2+𝐵2−𝐵

2 (3.14)

with 𝑘𝑦𝑓 = 𝑘𝑦𝑓′ + 𝑗𝑘𝑦𝑓

′′ , 𝐴 = 𝑘𝑥𝑛2 + 𝑘𝑧𝑚

2 − 𝑘𝑦𝑓′2 + 𝑘𝑦𝑓

′′2, 𝐵 = 2𝑘𝑦𝑓′ 𝑘𝑦𝑓

′′

The real part of (3.14) determines the phase velocity of the wave, and the imaginary part

characterizes its damping. The phase velocity of the wave, incident on the front wall of the

resonator, is determined by the following relation:

- 60 -

𝑣𝑝ℎ =𝑅𝑒{𝜔}

𝑘𝑧 (3.15)

This speed can be either more (at 𝑘𝑥𝑛2 > 𝑘𝑥0𝑛

2 ) or less (at 𝑘𝑥𝑛2 < 𝑘𝑥0𝑛

2 ) than the speed of

light, where

𝑘𝑥0𝑛2 = 𝑘𝑦𝑓

′2 − 𝑘𝑦𝑓′′2 − 𝑘𝑧𝑚

2 ± 2𝑘𝑧𝑚√2𝑘𝑦𝑓′ 𝑘𝑦𝑓

′′ + 𝑘𝑧𝑚2 (3.16)

Equality 𝑘𝑥𝑛2 = 𝑘𝑥0𝑛

2 (at which the wave has a phase velocity equal to the speed of light)

cannot be fulfilled exactly because 𝑘𝑦𝑓′2 , 𝑘𝑦𝑓

′′2 and 𝑘𝑧𝑚2 are discrete and fixed. The condition

for the greatest closeness of the phase velocity of a wave to the velocity of light, thus, is

the requirement 𝑚𝑖𝑛|𝑘𝑥𝑛2 −𝑘𝑥0𝑛

2 |. The mode (or set of modes) that satisfies this requirement

forms the main resonance arising as a result of the interactions in it of elementary free

oscillations.

3.3 Pure copper cavity. Measurements and interpretation

To check the accuracy of manufacturing, firstly an experimental resonances of pure

copper cavity are compared with theoretical results for the cavity.

The resonant behavior of pure copper cavity has been investigated experimentally

in CANDLE SRI by measuring the testing device (cavity) transmission S-parameter 𝑺𝟏𝟐

(forward voltage gain) [69] using Network Analyzer ZVB14 [70]. The measurements are

performed for the rectangular cross-section copper cavities.

Measured narrow-band extremes are compared with the standing wave

eigenfrequencies of rectangular cavity with perfectly conducting walls [72] with same

dimensions (𝑎′ = 22, 32, 42 𝑚𝑚, 𝑏 = 65𝑚𝑚, 𝑙 = 200 𝑚𝑚):

𝑓𝑚𝑛𝑞 =𝑐

2√(

𝑚−1

𝑎′)2

+(𝑛−1

𝑏)2

+ (𝑞−1

𝑙)2

𝑚, 𝑛, 𝑞 = 1,2,3… {

𝑖𝑓 𝑚 = 1, 𝑛, 𝑞 > 1 𝑖𝑓 𝑛 = 1, 𝑚, 𝑞 > 1𝑖𝑓 𝑞 = 1, 𝑚, 𝑛 > 1

(3.17)

Note that in IHLCC cavity when 𝑑 = 0, both equations (𝑃1 and 𝑃2) are transformed

to equation for the vertical wave number of the ideal resonator (Eq. 3.12): 𝑆𝑖𝑛ℎ(𝑎𝑘𝑦) = 0

with solution 𝑘𝑦 = 𝑗 𝜋(𝑞 − 1) 𝑎⁄ . Similarly, in the limiting case of 𝐾 = 𝑘𝑦 (휀′𝜇′ = 1) we have

- 61 -

both from (Eq. 3.10.a) and (Eq. 3.10.b): 𝑆𝑖𝑛ℎ(𝑎′𝑘𝑦) = 0 with solution 𝑘𝑦 = 𝑗 𝜋(𝑞 − 1) 𝑎′⁄ .

In these extreme cases, all three eigenvalues are independent from each other.

𝑎′ = 22 𝑐𝑚 𝑎′ = 32 𝑐𝑚

N m n q Exp. Calc. N m n q Exp. Calc.

1 2 2 6 8.04 8.116 1 1 1 12 8.25 8.25

2 2 2 6 8.12 8.116 2 1 4 8 8.7 8.69

3 2 2 7 8.48 8.489 3 1 1 13 9.06 9

4 2 2 8 8.86 8.909 4 3 1 2 9.4 9.405

5 1 2 13 9.31 9.291 5 1 3 12 9.46 9.453

6 2 2 9 9.39 9.371 6 3 2 2 9.69 9.684

7 2 1 10 9.56 9.594 7 2 3 11 9.97 9.976

8 1 4 10 9.65 9.670 8 3 2 6 10.36 10.358

9 1 5 6 9.92 9.963 9 3 3 5 10.88 10.872

10 1 2 14 10.04 10.019 10 3 2 8 10.97 10.99

11 2 4 5 10.15 10.169 11 1 6 1 11.58 11.539

12 1 5 7 10.28 10.269 12 1 6 3 11.65 11.636

13 1 1 15 10.51 10.50 13 3 4 3 11.76 11.750

14 1 5 8 10.6 10.619 14 1 6 5 11.91 11.922

15 1 5 9 11.01 11.009 15 1 4 14 11.96 11.960

16 1 3 15 11.46 11.470 16 3 3 9 12.05 12.050

17 1 6 5 11.93 11.922 17 1 3 16 12.16 12.160

18 1 5 12 12.3 12.38 18 3 3 10 12.43 12.440

19 2 1 15 12.53 12.519 19 3 2 12 12.7 12.700

20 2 5 8 12.605 12.62 20 3 1 13 13.0 12.996

21 2 2 15 12.71 12.73 21 2 2 17 13.11 13.088

22 1 5 13 12.9 12.892 22 3 4 10 13.48 13.468

23 2 1 16 13.14 13.155

24 2 3 15 13.33 13.343

25 2 6 3 13.47 13.486

26 2 4 6 13.57 13.59

27 2 5 11 13.7 13.709

28 2 1 17 13.8 13.802

29 2 3 16 13.94 13.941

30 1 5 15 13.98 13.980

Table 3.1.a: Comparison of the measured (Exp., GHz) and calculated (Calc., GHz) values of the

eigenfrequencies of a rectangular cavity with a height of 22 𝑐𝑚 and 32 𝑐𝑚.

Table 3.1 shows the correlation of eigenfrequencies of the cavity with the perfectly

conducting walls, calculated by Eq. 3.12, with measured peaks.

- 62 -

𝑎′ = 42 𝑐𝑚

N m n q Exp. Calc. N m n q Exp. Calc.

1 1 4 7 4.12 4.129 10 1 3 10 8.18 8.177

2 1 4 8 4.33 4.344 11 1 2 12 8.56 8.5666

3 1 5 2 4.65 4.63 12 2 1 12 8.98 8.9899

4 1 2 7 5.04 5.057 13 1 4 9 9.17 9.1601

5 1 3 5 5.52 5.505 14 1 3 12 9.45 9.4533

6 1 3 16 6.08 6.080 15 2 5 2 9.93 9.9260

7 2 3 5 6.66 6.562 16 3 3 9 10.42 10.408

1 1 10 6.75 17 3 3 10 10.84 10.857

8 1 4 5 7.55 7.545 18 3 4 7 10.90 10.918

9 2 3 8 7.85 7.850

Table 3.1.b: Comparison of the measured (Exp., GHz) and calculated (Calc., GHz) values of the

eigenfrequencies of a rectangular cavity with 42 𝑐𝑚 height.

As can be seen from Table 3.1, there is a good matching (up to the second decimal)

between experimentally obtained and calculated eigenvalues. This is achieved due to the

high precision machining of internal surfaces of the cavity with keeping dimensional

accuracy.

The main result of the above research is the establishment of agreement of

experimentally obtained narrowband peaks with resonant frequencies of the rectangular

copper cavities. This means that the accuracy of manufacturing of copper resonator

component allow to obtain the corresponding frequency distribution for the IHLCC cavity

with acceptable distortions.

3.4 IHLCC cavity resonance frequency comparison with experimental results

The transmission 𝑺𝟏𝟐 parameters of IHLCC cavity has been measured to perform a

comparison between resonant behaviors of real structure and IHLCC.

The defining parameters of the cavity are the following: the height of the inner

chamber 𝑎, the thickness of the low conductive layer 𝑑 =𝑎′−𝑎

2, the width 𝑏 and length 𝑙 of

the chamber, conductivity 𝜎1 and relative dielectric constant 휀1 of low conductive layer.

The choice of material and the thickness of a low-conductive metallic layer,

covering the top and bottom walls of cavity is conditioned by following two factors:

- 63 -

1. network analyzer ZVB14 provides spectral distribution in the range of 0 − 14 𝐺𝐻𝑧.

The resonance frequency is determined roughly by the geometrical parameters of

the cross-section of the cavity: 𝑓𝑟𝑒𝑧 ≈𝑐

2𝜋√2/𝑎𝑑, which depends on the thickness of

the inner layer 𝑑 and structure height 𝑎. Having a bunch in a structure imposes the

lower limit on its height (𝑎~1 𝑐𝑚). Also with the increase of the height the impedance

decreases, so the height should be near to its lower limit. For a structure with 2 −

4 𝑐𝑚 height the thickness of the coating should be about 1 − 0.5 𝑚𝑚 to have a

resonance frequency in 0 − 14 𝐺𝐻𝑧 range, given by measuring devise.

2. for the double-layered parallel plates the condition of resonance 휁 ≈ 𝑍0𝜎1𝑑~1 is

derived in chapter 2. Assuming that for the resonator the condition should be

approximately the same, the low conducting material is chosen to have 𝜎1~2.65 𝑆𝑚⁄

conductivity for a layer with 𝑑 = 1 𝑚𝑚 thickness.

As a result, germanium (Ge) crystal plates with sizes 𝑙 = 30 c𝑚, 𝑏 = 6 c𝑚, 𝑑 =

1 𝑚𝑚 with conductivity 𝜎1 = 2 Ω−1𝑚−1 [73] and the relative dielectric constant 휀1 = 16,

were selected. Measurements were carried out for three different heights: 𝑎 = 2 𝑐𝑚, 𝑎 =

3 𝑐𝑚 and 𝑎 = 4 𝑐𝑚 (𝑎′ = 2.2, 𝑎′ = 3.2 and 𝑎′ = 4.2 𝑐𝑚, respectively).

For the above-given vertical apertures and the Ge layer thickness, the skin depth at

resonant frequency 𝛿 is larger than the Ge layer thickness by factor of 3 (𝛿/𝑑 > 3). The

predicted resonant frequencies of the corresponding two parallel plates lie in the X-Band

region between 9 − 15 𝐺𝐻𝑧.

3.4.1 Comparison with flat two-layer structure

The measured resonant frequencies of the test cavity with laminated horizontal

walls have been compared with resonant frequencies of the corresponding two-

dimensional two-layered structure (outer layer is assumed as a perfect conductor) with

inner 1 𝑚𝑚 Ge layers.

The longitudinal impedance of two parallel infinite laminated plates is derived in the

second chapter of the dissertation. For an ultrarelativistic particle moving parallel to the

plates along geometrical axis of the structure with perfectly conducting outer and low

- 64 -

conducting inner metallic layers the longitudinal impedance can be presented as (Eq. 2.15,

Eq. 2.33, Eq. 2.42 and Eq. 2.43)

𝑍∥ =𝑍0

16𝜋2∫

𝑘𝑢(𝑝2−𝑢2)𝑆ℎ(2𝑝𝑑)

𝑆1𝑆2

∞

−∞𝑑𝑢 (3.18)

where

𝑆1 = 𝑝𝐶ℎ(𝑝𝑑)𝐶ℎ(𝑢𝑎) + 𝑢𝑆ℎ(𝑝𝑑)𝑆ℎ(𝑢𝑎)

𝑆2 = (𝑢2 + 𝑘2 − 𝑝2)𝐶ℎ(𝑝𝑑)𝐶ℎ(𝑢𝑎) +

𝐾𝑘2

𝑞𝑆ℎ(𝑝𝑑)𝑆ℎ(𝑢𝑎)

(3.19)

with 𝑎1 = 𝑎 2⁄ , 𝑘 = 𝜔 𝑐⁄ longitudinal wavenumber and

𝐾 = √𝑢2 + 𝑘2(1 − 휀1) − 𝑗𝑘𝑍0𝜎 (3.20)

Figure 3.2 presents the longitudinal impedance curves for such parallel laminated

plates with 2 𝑐𝑚, 3 𝑐𝑚 and 4 𝑐𝑚 vertical apertures. The distances between the plates and

material of the inner layers (germanium) are the same as in the tested cavities. The

dashed lines indicate the measured main resonant frequencies of the corresponding

copper cavities with laminated top and bottom walls.

Figure 3.2: The longitudinal impedance of perfect conductor-Ge infinite parallel plates

(solid lines) and measured resonant frequencies of copper cavity with Ge inner layers at

the top and bottom walls (dashed lines). The Ge layer thickness is 1 𝑚𝑚, the vertical

apertures are 2 𝑐𝑚, 3 𝑐𝑚 and 4 𝑐𝑚.

- 65 -

Figure 3.2 shows that there is a good agreement between measured and calculated

resonant frequencies. The relative difference in resonant frequency is 2.6 % for 2 𝑐𝑚 ,

3.5 % for 3 𝑐𝑚 and 7 % for 4 𝑐𝑚 aperture cavities. The difference is caused mainly due to

the limited volume of the cavity: the presence of highly conductive sidewalls. The presence

of additional bursts is the result of the display of modal structure of the test cavity. Their

contribution is weakened by decreasing of the distance between plates. Thus, by

decreasing the height of cavity, its resonant characteristics approach to the characteristics

of a flat structure consisting of two identical two-layer parallel plates.

3.4.2 Comparison with IHLCC resonator theoretical results

Resonant frequencies of the cavity with outer perfectly conducting walls and inner

germanium layers at the top and bottom walls can be obtained from Eq. 3.11.

Experimentally fixed resonances (Figure 3.5) can be identified as concrete solutions

(modes) of dispersion relations given by Eq. 3.11. Choosing the needed mode (𝑘𝑥, 𝑘𝑧) the

main resonances frequencies can be obtained for our IHLCC cavity with ideally conductive

walls. Table. 3.2 shows comparison of experimentally observed resonance frequencies of

the test cavity (“Experiment” in table) and analytically calculated resonances of cavity with

outer perfectly conducting walls (“Calculated” in table).

Aperture

2

Experiment

4

5

Calculated Rel. dif.

2 𝑐𝑚

12.32 𝐺𝐻𝑧

12.29 𝐺𝐻𝑧

0.2 %

3 𝑐𝑚 11.42 𝐺𝐻𝑧

11.36 𝐺𝐻𝑧

0.5 %

4 𝑐𝑚

10.6 𝐺𝐻𝑧

10.49 𝐺𝐻𝑧

1 %

Table. 3.2: The main resonances obtained experimentally and calculated for a cavity with

perfectly conducting walls and inner Ge layers at the top and bottom walls.

0.251

0.339

0.754

1.001

1.257

1.508

1.760

As can be seen from Table. 3.2, there is a good agreement on measured and

calculated results. The relative differences between the calculated and measured

frequencies (“Rel. dif.” in table) are below 1 %.

As it was noted earlier (see section 3.2), free electromagnetic waves formed in the

resonator form standing waves, which can be represented as the superimposition of two

oppositely directed traveling waves: incident on the front wall of the resonator and going in

- 66 -

the opposite direction (see Eq.3.13). The phase velocities of these waves are equal in

absolute value and opposite in sign (since they propagate in opposite directions): 𝑣𝑝ℎ =

𝑐 𝑅𝑒{𝑘} 𝑘𝑧⁄ with 𝑘 =𝜔

𝑐= √𝑘𝑥2 + 𝑘𝑧2 − 𝑘𝑦2 ; 𝑘𝑥 and 𝑘𝑧 are real numbers determined by the

indices m and n and 𝑘𝑦 is determined by the equations (3.10) and is a set of complex

numbers. Thus, k is a complex number, the real part of which determines a discrete series

of eigenfrequencies of the resonator, and its imaginary part, caused by the finite

conductivity of the inner layers covering the inside walls, determines the finite Q of the

resonator at its eigen frequencies. Determining k from the equations (3.10), we can

determine the dependence of the phase velocity of the partial traveling waves in the cavity,

depending on the frequency. As noted earlier, the exact equality of the phase velocity of

the partial wave in the resonator with the speed of light is unlikely. Investigations of the

solutions of the equations (3.10) show that in the frequency range 0-14 GHz with the

corresponding cavity parameters, only equation 𝑃4 = 0 (3.10d) give solutions for slow

partial waves. The corresponding graphs are shown in the Figure 3.3.

Figure 3.3: The phase velocities of the partial waves in the resonator for modes 𝑃4,𝑛𝑓𝑚 (solutions of

equation 𝑃4 = 0) for 𝑛 = 2, 𝑓 = 1,2, 𝑚 = 1,2,3,4,…; 𝑎 = 1 𝑐𝑚 (left), 1.5 𝑐𝑚 (middle), 𝑎 = 2 𝑐𝑚 (right).

Each of the graphs shows two sequences (discrete curves) of phase velocities of

the modes: 𝑃4;21𝑚 and 𝑃4;2,2𝑚 with monotonically decreasing phase velocities. On all three

graphs, a discrete curve depicting a sequence of modes 𝑃4;2,2𝑚 intersects the

synchronization line 𝑣𝑝ℎ 𝑐⁄ = 1. The second discrete curve containing the phase velocities

of the modes 𝑃4;21𝑚 is located above the line 𝑣𝑝ℎ 𝑐⁄ = 1, gradually approaching it as the

frequency increases and intersecting with the first line. In the first and second cases

(for 𝑎 = 1𝑐𝑚 and 𝑎 = 1.5𝑐𝑚 ), the measured fundamental resonance frequency

corresponds to the intersection point of two discrete lines adjacent to the synchronization

- 67 -

line and calculated (for two – dimensional planar case) is closed to intersection point of

first curve and line of synchronization. In the third case, the situation is reversed: the

measured fundamental resonance frequency corresponds to intersection point of first

curve and line of synchronization and calculated is closed to the intersection point of two

discrete lines.

It is obvious that in all three cases the main resonance is formed by several modes

with closed frequencies and close in phase velocity: modes 𝑃4;2,1,15,

𝑃4;2,1,16, 𝑃4;2,2,15, 𝑃4;2,2,16, 𝑃4;2,2,17 for 𝑎 = 1𝑐 , 𝑃4;2,1,13, 𝑃4;2,1,14, 𝑃4;2,1,15, 𝑃4;2,213, 𝑃4;2,2,14, 𝑃4;2,2,15

for 𝑎 = 1.5𝑐𝑚 and 𝑃4;2,1,13, 𝑃4;2,1,14, 𝑃4;2,1,15, 𝑃4;2,213, 𝑃4;2,2,14, 𝑃4;2,2,15 for 𝑎 = 2𝑐𝑚.

3.5 Summary

The chapter is the logical follow-up of the study of high frequency single-mode

accelerating structures with laminated metallic walls (flat and cylindrical) [50, 75, 76].

Theoretical model of a rectangular cavity with horizontal double layer metallic walls

is created and electro-dynamic properties of the model are studied. The model with

sufficient accuracy displays experimental tested device and enables clearly interpret the

measurement results. Extreme frequencies on the experimental curves have been

correlated with the respective modes of resonator model. The experimental results are

also compared with theoretical results for two infinite parallel plates. A good agreement

between them stimulates the experimental study of beam radiation and acceleration in

laminated structures. The corresponding experimental program at AREAL test facility is

foreseen.

- 68 -

Chapter 4: Electron bunch compression in single-mode

structure

4.1 Introduction

The generation and acceleration of ultrashort electron bunches is an important

issue for generation of coherent radiation in THz and Infrared regions, driving the free

electron lasers (FEL) and direct applications for ultrafast electron diffraction [14-16, 77-79].

Length of bunches generated in RF guns is limited due to technical characteristics

of photocathode and lasers. So for having a sub-ps bunches one need to shorten ps

bunches generated in RF gun. There are several methods for obtaining sub-ps bunches,

e.g. magnetic chicanes are used for high energy beams shortening [80], velocity and

ballistic bunching are for low energy beams [26-29]. For all these methods one needs to

have an energy modulated beam.

It is well known that the relativistic charge, moving along the structure, interacts with

the surrounding environment and excites the electromagnetic field known as wakefield

[36-38, 59-61]. Structures are usually the disk, plasma or dielectric loaded channels that

drive the corresponding wakefield accelerator (WFA) concepts [25, 34, 35, 37, 55, 57], the

performance of which can be improved by using the asymmetric driving beam to achieve

high transformer ratio [45, 83]. For the charge distribution the wakefield acts back to

bunch particles producing an energy modulation within the bunch by longitudinal wake

potential. The form of longitudinal wake potential depends both on the bunch charge

distribution and the surrounding structure [36]. Although for Gaussian bunch the energy

modulation is driven by the bunch head-tail energy exchange, for rectangular and

parabolic bunch shapes the energy modulation at high frequency can be obtained.

To have an energy modulation disc and dielectric loaded structures are used.

These kind of structures have high order modes excited by a charge when it passes the

structure, which leads to energy storage in that parasitic modes and beam instabilities [35-

46]. To avoid such effects one needs a single-mode structure. In this chapter wakefield

induced bunch energy modulation during bunch interaction with the single-mode

- 69 -

structures is studied. As a single mode structure cold plasma and recently proposed

internally coated metallic tube (ICMT) are considered [47-50]. In a cold plasma purely

sine-like wakefield without damping are generated, while in ICMT wakefields are sine-like

with damping factor [50].

The direct application of the holding integral for bunch rectangular or parabolic

shape and point wake potential lead to energy modulation within the bunch at the excited

mode frequency. To obtain sub-𝑝𝑠 bunches, a structure with excited frequency higher

than 𝑇𝐻𝑧 is needed. In cold plasma and ICMT, for appropriate parameters, the excited

mode is within sub-𝑇𝐻𝑧 to 𝑇𝐻𝑧 frequency region.

For comparatively low energy bunches wakefield generated energy modulation

causes velocity modulation and for proper ballistic distance leads to density modulation or

microbunching. To have that effect in proper distance, energy modulation in a bunch after

the structure should be greater than several 𝑘𝑒𝑉, which can be achieved with 100 𝜇𝑚 −

500 μ𝑚 bunches. Electron bunches with 100 𝑝𝐶 charge and 10 𝑀𝑒𝑉 energy are observed,

which corresponds to usual values in modern linear accelerators [84].

4.2 Wakefields and Impedances

A relativistic charge 𝑄 , passing throw a structure interacts with it. Due to this

interaction the particle radiates electromagnetic fields, known as wakefields, which then

can act to the test charge 𝑞. From the causality principle, wakefields generated by an

ultrarelativistic driving charge should vanish in front of it [37]. The longitudinal component

of the wakefield produces energy gain (or loss) of the test particle, while the transverse

component produces the transverse kick of the test particle.

- 70 -

Figure 4.1: Driving 𝑄 and test 𝑞 charge moving in a structure

The longitudinal wake function (point wake potential) is the integrated longitudinal

Lorentz force excited in a structure by a point charge 𝑄 experienced by the test particle

traveling on an 𝑠 distance behind it [36]

𝑤𝑧(𝒓, 𝒓𝑸, 𝑠) = −1

𝑄∫ 𝐸𝑧 (𝒓, 𝒓𝑸, 𝑧, 𝑡 =

𝑧+𝑠

𝑐)𝑑𝑧

∞

−∞ (4.1)

where 𝐸𝑧 is the longitudinal component of excited electric field, 𝒓 is a transverse offset of

test particle, 𝒓𝑸 is a transverse offset of source particle and 𝑐 is the speed of light in

vacuum (Figure 4.1). In the definition 𝑠 > 0 corresponds to the distance behind the driving

point charge, 𝑠 = 0 to the position of the charge and 𝑠 < 0 to the distance in front of the

charge.

The transverse wake function is defined as an integrated transverse Lorentz force

acting on a test particle

𝒘⊥(𝒓, 𝒓𝑸, 𝑠) = −1

𝑄∫ (𝑬 + 𝒗 × 𝑩)⊥ (𝒓, 𝒓𝑸, 𝑧, 𝑡 =

𝑧+𝑠

𝑐) 𝑑𝑧

∞

−∞ (4.2)

For a longitudinal distributed 𝜆(𝑠) driving bunch, the wake potentials are defined as

an integrated longitudinal and transverse Lorentz forces excited by a bunch at the

position of the test particle traveling on an 𝑠 distance behind it. Here distance 𝑠 is

- 71 -

measured from the center of driving bunch. The longitudinal wake potential could be

obtained from the wake functions using the convolution integral

𝑊∥(𝑠) = ∫ 𝑤𝑧(𝑠 − 𝑠′)𝜆(𝑠) 𝑑𝑠′𝑠

−∞ (4.3)

here integration is from −∞ to 𝑠, because there is no field in front of driving bunch.

The total energy loss of the driving charge in a structure due to its radiation is

proportional to the square of the charge

Δ𝑈 = 𝑄2𝐾𝑙𝑜𝑠𝑠 (4.4)

The proportionality factor 𝐾𝑙𝑜𝑠𝑠 is a longitudinal loss factor and can be presented by

wake potentials

𝑘𝑙𝑜𝑠𝑠 = ∫ 𝑊∥(𝜏)𝜆(𝜏)𝑑𝜏∞

−∞ (4.5)

The wake potential is a function of time, while Fourier transformation of the wake

potential gives an information in frequency domain and is called the impedance [36, 47]

𝑍∥(𝜔) = ∫ 𝑊𝑧(𝜏)𝑒−𝑖𝜔𝜏𝑑𝜏

∞

−∞=1

𝑐∫ 𝑊𝑧(𝑧)𝑒

−𝑖𝜔𝑧𝑐⁄ 𝑑𝑧

∞

−∞

𝑊𝑧(𝜏) =1

2𝜋∫ 𝑍∥(𝜔)𝑒

𝑖𝜔𝜏𝑑𝜔∞

−∞

(4.6)

4.3 Ballistic bunching

Traveling in the structure bunch particles gain (loss) energy due to interaction with

wakefields generated by themselves and particles in front of them. Ballistic bunching

method of studying microbunching and bunch compression is used. In ballistic method it is

assumed that in the structure charge is rigid and only energy modulations occur. The

energy modulation leads to particles velocity change in the structure and for non-

ultrarelativistic bunches charge density redistribution occurs in drift space after the

structure (Figure 4.2). For appropriate initial bunch energy and energy spread after the

structure, bunches with several 𝜇𝑚 length can be formed.

- 72 -

Figure 4.2: Geometry of ballistic bunching (blue line illustrates the wakefield)

After bunch passes the structure energy gain (loss) of 𝑖-th particle with initial 𝑠𝑖

coordinate is determined by

Δ𝑈(𝑠𝑖) = 𝑞𝑄𝑊𝑧(𝑠𝑖)𝐿 (4.7)

where 𝑞 is a charge of test particle, 𝑄 is a total charge of a source bunch, 𝑊𝑧(𝑠𝑖) is a wake

potential in a structure and 𝐿 is a structure length.

Velocity of 𝑖-th particle on the entrance of drift space can be presented as

𝛽(𝑠𝑖) = √1 − (𝑈𝑟𝑒𝑠𝑡

𝑈𝑓(𝑠𝑖))2

(4.8)

where 𝑈𝑟𝑒𝑠𝑡 is a particle rest energy and 𝑈𝑓(𝑠𝑖) = 𝑈0 + Δ𝑈(𝑠𝑖) is an energy of 𝑖-th particle

after the structure. Here cold bunch approximation is assumed, which means that there is

no energy spread in the initial bunch. Having a velocity of each particle at the end of the

structure, the coordinate 𝑠𝑖,𝑓 of 𝑖-th particle after passing 𝑙𝑑𝑟𝑖𝑓𝑡 distance in a drift space can

be calculating by

𝑠𝑖,𝑓 = 𝑠𝑖 + Δ𝑠𝑖 = 𝑠𝑖 + 𝑙𝑑𝑟𝑖𝑓𝑡 (1 −𝛽(𝑠𝑖)

𝛽0) (4.9)

- 73 -

where 𝑠𝑖 is the initial coordinate of 𝑖 -th particle, 𝛽0 = √1 − (𝑈𝑟𝑒𝑠𝑡

𝑈0)2

is the bunch initial

velocity, 𝛽(𝑠𝑖) is the velocity of 𝑖-th particle after passing the structure. With the coordinate

of each particle in a bunch after a drift space, one can obtain the bunch’s shape.

Numerical simulations are done for observing wakefield effects on various line

charge distributions. The bunch is presented as an ensemble of 𝑁 particles that

experience the energy deviations due to longitudinal wakefield. For simulations 𝑁 = 106

particles have been used.

4.4 Energy modulation, bunch compression and microbunching in cold

neutral plasma

Bunch charge density modulation in infinite cold neutral plasma is studied via

ballistic bunching method. It is assumed that the electrons temperature is much higher

than the ions temperature (cold plasma). Under this assumption the plasma can be

considered as a medium with 휀 = 1 −𝜔𝑝2

𝜔2⁄ macroscopic dielectric constant, where the

plasma frequency is 𝜔𝑝 = 56.4 ∗ 103√𝑛𝑒 𝐻𝑧, with 𝑛𝑒 being the plasma electron density (in

𝑐𝑚−3 units) [47]. The wake function of the relativistic point charge, passing the cold neutral

plasma, is given by [47, 48]

𝑤𝑧(𝑠) = −2𝐻(𝑠)𝐾𝑙𝑜𝑠𝑠 cos(𝑘𝑝𝑠) (4.10)

where 𝐻(𝑠) is the Heaviside step function, 𝑘𝑝 =𝜔𝑝

𝑐⁄ is the wave number, 𝐾𝑙𝑜𝑠𝑠 is the

longitudinal loss factor of plasma channel.

- 74 -

Figure 4.3: Point charge wake function in cold neutral plasma (𝜔𝑝 = 56.4 ∗ 103√𝑛𝑒 𝐻𝑧 and

𝐾𝑙𝑜𝑠𝑠 = 45𝑘𝑉

𝑚/𝑝𝐶)

Numerical simulations of bunch compression and microbunching have been

performed for the driving bunch with 100 𝑝𝐶 charge and 10 𝑀𝑒𝑉 initial energy.

Wake potential generated by a Gaussian bunch in plasma channel can be obtained

by Eq. 4.2

𝑊∥(𝑠) =1

√2𝜋𝜎∫ 𝑤𝑧(𝑠 − 𝑠′)𝑒

−𝑠2

2𝜎2 𝑑𝑠′𝑠

−∞ (4.11)

Figure 4.4 presents the wake potential of the Gaussian charge distribution with

0.1 𝑚𝑚 rms bunch length in plasma channel with 𝑓𝑝 =𝜔𝑝

2𝜋⁄ = 1.43 𝑇𝐻𝑧 plasma frequency

(wavelength 𝜆𝑝 = 0.2 𝑚𝑚).

- 75 -

Figure 4.4: Gaussian bunch wake potential.

Figure 4.4 shows that the wake potential is decelerating at bunch’s head (𝑠 < 0)

and accelerating at bunch’s tail (𝑠 > 0). This leads to bunch compression till some critical

drift space distance, after which particles with coordinates 𝑠 > 0 overtake particles with

coordinates 𝑠 < 0. For distances bigger than the critical distance bunch starts to expand

and debunching process starts. From Figure 4.4 it can be seen that the central part

(~70%) of the bunch sees focusing wake, while bunch’s tail sees a decelerating wake.

Compressed bunch shape at various drift space distances after 2 𝑚 plasma channel

is shown in Figure 4.5. The dashed line represents the initial Gaussian bunch distribution.

It can be seen, that till 3.5 𝑚 drift space length bunch compresses, after which it starts to

expand. On 3.5 𝑚 drift space length the rms length of bunch’s central part that contains

70 % of the particles is 15 𝜇𝑚.

- 76 -

a

b

c

d

Figure 4.5: Line charge distribution after bunch passes 𝑙𝑑𝑟𝑖𝑓𝑡 distance in drift space

𝑎) 𝑙𝑑𝑟𝑖𝑓𝑡 = 1.5 𝑚, b) 𝑙𝑑𝑟𝑖𝑓𝑡 = 3.5 𝑚, c) 𝑙𝑑𝑟𝑖𝑓𝑡 = 4 𝑚, d) 𝑙𝑑𝑟𝑖𝑓𝑡 = 4.5 𝑚

To find the maximum compression, rms length dependence of distance is

calculated. rms length of the Gaussian bunch 70 % (central part) and 100 % particles

versus bunch travel distance in drift space is shown in Figure 4.6. As it was mentioned,

bunch’s tail sees defocusing wakefield. This defocused part artificially increases rms

length of a whole bunch and it always stays above 70 𝜇𝑚 . For 70 % bunch particles

maximum bunch compression is reached at 4 𝑚 distance (bunch rms length is ~ 12 𝜇𝑚)

after which the debunching process starts.

- 77 -

Figure 4.6: The Gaussian bunch longitudinal rms length for 70 % (left) and 100 % (right)

versus drift distance.

To have a microbunching, one needs a wakefield with several times smaller wave

length than bunch’s length. High frequency fields generated by a Gaussian distribution

are very low, because of exponential damping factor in distribution. While for Gaussian

bunch it is not possible to have a microbunching in appropriate distances, for some non-

Gaussian distributed bunches it is possible.

For the rectangular uniform charge distribution the wake potential in plasma

channel produces bunch energy modulation at the excited mode wavelength.

Rectangular uniform line charge distribution is in the form of

𝜆(𝑠) = { 1

2𝑎, |𝑠| ≤ 𝑎

0 , |𝑠| > 𝑎 (4.12)

and the rms length of the bunch is given by 𝜎 =𝑎

√3.

Wake potential of such a bunch can be derived from Eq. 4.2

𝑊𝑧(𝑠) = ∫ 𝑤𝑧 (𝑘𝑝(𝑠 − 𝑠′)) 𝜆(𝑠′)𝑑𝑠′

∞

−∞= {

0 , 𝑠 < −𝑎

𝑘𝑙𝑜𝑠𝑠

2𝑎𝑘sin (𝑘𝑝(𝑠 + 𝑎)) , |𝑠| ≤ 𝑎

𝑘𝑙𝑜𝑠𝑠

𝑎𝑘sin(𝑘𝑝𝑎) cos(𝑘𝑝𝑎) , 𝑠 > 𝑎

(4.13)

Figure 4.7 shows the wake potential generated by a rectangular uniform distributed

bunch of 0.58 𝑚𝑚 rms length (2 𝑚𝑚 total length) in a plasma channel with 0.6 𝑚𝑚 (𝑓𝑝 ≈

0.5 𝑇𝐻𝑧) excited mode wavelength.

- 78 -

Figure 4.7: Uniform bunch wake potential.

The energy modulation at the wakefield excited mode frequency leads to the charge

density modulation as the beam travels in free space.

a

b

c

d

Figure 4.8: Line charge distribution after passing 𝑙𝑑𝑟𝑖𝑓𝑡 distance in a drift space

𝑎) 𝑙𝑑𝑟𝑖𝑓𝑡 = 1.5 𝑚, b) 𝑙𝑑𝑟𝑖𝑓𝑡 = 2.5 𝑚, c) 𝑙𝑑𝑟𝑖𝑓𝑡 = 3.5 𝑚, d) 𝑙𝑑𝑟𝑖𝑓𝑡 = 5 𝑚

- 79 -

Charge density modulation at various drift space distances after 0.3 𝑚 plasma

channel is shown in Figure 4.8. As it can be seen, the density modulation leads to three

microbunches formation.

The best bunching is at the 3.5 𝑚 drift space distance, where the initial rectangular

bunch is transformed into three microbunches with 14 𝑝𝐶 charge and 40 𝜇𝑚 full width at

half maximum (FWHM). After that distance debunching process starts (Figure 4.8(d)).

Parabolic line charge distribution is taken in the form of

𝜆(𝑠) = { 1

2𝑎3(𝑎2 − 𝑠2), |𝑠| ≤ 𝑎

0 , |𝑠| > 𝑎 (4.14)

and rms length of the bunch is determined by 𝜎 =𝑎

√5.

The longitudinal wake potential of parabolic charge can be derived from Eq. 1.2

𝑊𝑧(𝑠) = ∫ 𝑤𝑧(𝑘(𝑠 − 𝑠′))𝜆(𝑠′)𝑑𝑠′

∞

−∞=

{

0 , 𝑠 < −𝑎

𝑘𝑙𝑜𝑠𝑠

𝑎3𝑘3[sin(𝑘(𝑠 + 𝑎)) − 𝑘𝑎 cos(𝑘(𝑠 + 𝑎)) − 𝑘𝑠] , |𝑠| ≤ 𝑎

𝑘𝑙𝑜𝑠𝑠

𝑎3𝑘3[sin(𝑘(𝑎 + 𝑠)) + sin(𝑘(𝑎 − 𝑠)) − 𝑘𝑎(cos(𝑘(𝑎 + 𝑠)) + cos(𝑘(𝑎 − 𝑠)))] , 𝑠 > 𝑎

(4.15)

The wake potential for parabolic charge distribution of 0.45 𝑚𝑚 rms length (2 𝑚𝑚

full length) is shown in Figure 4.9. Dashed line represents the bunch shape. To satisfy a

microbunching criterion a plasma channel with 0.6 𝑚𝑚 ( 𝑓𝑝 ≈ 0.5 𝑇𝐻𝑧 ) excited mode

wavelength is taken.

- 80 -

Figure 4.9: Parabolic bunch wake potential.

Particles energy modulation is observed with linear ramp of the average energy

deviation.

Figure 4.10 shows the parabolic charge density modulation at various drift space

distances after 1 𝑚 plasma channel length. The density modulation leads to microbunching

and initial bunch transforms into three microbunches.

a

b

c

d

Figure 4.10: Line charge distribution after passing 𝑙𝑑𝑟𝑖𝑓𝑡 distance in a drift space:

𝑎) 𝑙𝑑𝑟𝑖𝑓𝑡 = 1 𝑚, b) 𝑙𝑑𝑟𝑖𝑓𝑡 = 3 𝑚, c) 𝑙𝑑𝑟𝑖𝑓𝑡 = 5 𝑚, d) 𝑙𝑑𝑟𝑖𝑓𝑡 = 7 𝑚

The best bunching is observed on a 5 𝑚 drift space length, where microbunches

with 10 𝑝𝐶 charge 20 𝜇𝑚 length (FWHM) are formed. It can be seen that at 7 𝑚 drift space

length debunching of all three microbunches occurs.

- 81 -

4.5 Energy modulation, bunch compression and microbunching in internally

coated metallic tube

Energy and charge density modulation of Gaussian, rectangular uniform and

parabolic charge distributions in ICMT are observed. The geometry of ICMT is shown in

Figure 4.11.

Figure 4.11: Geometry of the ICMT

In a high frequency range, for a thin, low conductivity inner layer and infinite thick,

perfectly conducting outer layer the ICMT is a single-mode structure with 𝑓0 =𝑐

2𝜋√2 𝑎𝑑⁄

resonant frequency, where 𝑎 is the inner radii of a layer and 𝑑 is its thickness.

Wake function generated in ICMT by an ultrarelativistic point charge moving along

the structure axis can be presented as [50]

𝑤∥(𝑠) = −𝑍0𝑐

𝜋𝑎2𝑒−𝛼𝑠 [cos(𝑘𝑎𝑠) −

𝛼

𝑘𝑎sin(𝑘𝑎𝑠)] (4.17)

where, 𝑘𝑎 = √𝑘02 − 𝛼2 , 𝑘0 =

2𝜋

𝑐𝑓0 and 𝛼 damping factor is a function of layer thickness

and conductivity.

- 82 -

For non-ultrarelativistic particle (10 𝑀𝑒𝑉) the longitudinal wake function calculated in

[85] is used. Wake functions of a point charge with various energies are shown in Figure

4.12.

a

b

c

d

Figure 4.12: Wake functions of a point charge with gamma factor 𝛾 a) 𝛾 = 10, b) 𝛾 = 20, c) 𝛾 = 40, d) 𝛾 = 80

Matching the inner radii of the internal cover and its thickness one can obtain a

resonant frequency in 𝑇𝐻𝑧 region in the ICMT.

Numerical simulations are performed to determine bunch shape after drift space for

driving bunch charge of 𝑞 = 100 𝑝𝐶 with 𝑈0 = 10 𝑀𝑒𝑉 initial energy. The inner diameter of

a structure is 0.5 𝑚𝑚, electric conductivity of the first layer is 104 𝑆

𝑚, thickness of the first

layer is 1 𝜇𝑚. Excited mode frequency in ICMT with these parameters is approximately

𝑓0 =𝑐

2𝜋√2 𝑎𝑑⁄ = 3 𝑇𝐻𝑧 , which corresponds to 0.1 𝑚𝑚 wavelength. Point wake function of

such a structure is shown in Figure 4.12.b.

- 83 -

Figure 4.13 presents the wake potential of the Gaussian charge distribution with

0.1 𝑚𝑚 rms length (dashed line shows bunch shape).

Figure 4.13: 100 𝜇𝑚 Gaussian bunch wake potential.

The wake potential is decelerating at bunch’s head (𝑠 < 0) and accelerating at

bunch’s tail (𝑠 > 0). This energy modulation can lead to bunch compression for appropriate

drift space length.

Compressed bunch shape at various drift space distances after 1 𝑚 ICMT is shown

in Figure 4.14. Dashed line presents the initial Gaussian bunch distribution. The bunch

compresses till about 4 𝑚 drift space length after which tail of the bunch passes its head

and bunch starts to expand.

- 84 -

a

b

c

d

Figure 4.14: Line charge distribution at 𝑙𝑑𝑟𝑖𝑓𝑡 drift space distances.

𝑎) 𝑙𝑑𝑟𝑖𝑓𝑡 = 1 𝑚, b) 𝑙𝑑𝑟𝑖𝑓𝑡 = 2 𝑚, c) 𝑙𝑑𝑟𝑖𝑓𝑡 = 4 𝑚, d) 𝑙𝑑𝑟𝑖𝑓𝑡 = 6 𝑚

In ICMT, like in plasma, the Gaussian bunch sees strong focusing field on its central

part, while in its head and tail there is a weak field. Note that in ICMT there is no

debunching field on bunch’s tail. rms lengths of 70 % (central part) and 100 % particles

versus bunch travel distance in drift space is shown in Figure 4.15. The best compression

is reached at 3.8 𝑚 drift space distance, where a bunch with 70 𝑝𝐶 charge and 8.02 𝜇𝑚

rms length is formed.

- 85 -

Figure 4.15: rms of 70% (left) and 100% (right) of particles in a bunch versus drift space

distance.

To have a microbunching process in a structure one need a bunch several times

longer than the excited wavelength (0.1 𝑚𝑚 ). This criterion is taken into account for

studying microbunching of uniform rectangular and parabolic bunches.

Microbunching of rectangular uniform charge distribution with 0.17 𝑚𝑚 rms length

(0.6 𝑚𝑚 total length) in ICMT is studied. Wake potential of the bunch is presented in

Figure 4.16 (dashed line shows bunch shape). This wake potential generates an energy

modulation within the bunch at excited mode wavelength.

Figure 4.16: Uniform bunch wake potential.

The excited energy modulation leads to charge density modulation while it travels in

a free space. Figure 4.17 shows charge density modulations in various drift space

distances after 0.2 𝑚 ICMT. The initial uniform bunch transforms into several

microbunches.

- 86 -

a

b

c

d

Figure 4.17: Line charge distribution after passing 𝑙𝑑𝑟𝑖𝑓𝑡 distance in a drift space

𝑎) 𝑙𝑑𝑟𝑖𝑓𝑡 = 1 𝑚, b) 𝑙𝑑𝑟𝑖𝑓𝑡 = 2 𝑚, c) 𝑙𝑑𝑟𝑖𝑓𝑡 = 3 𝑚, d) 𝑙𝑑𝑟𝑖𝑓𝑡 = 4 𝑚

The best bunching is on 3 𝑚 drift space length, after which a debunching process

starts. Full width at half maximum (FWHM) of the first microbunch is 8 𝜇𝑚, which is a good

length for generating THz radiation. 7.8 % of total charge is concentrated in a full width

region of the first microbunch, and 10 % in three full width region. Full width of the second

microbunch is 16 𝜇𝑚 . 5.94 % of total charge is concentrated in a full width region and

10.5 % in three full width region.

Wake potential of parabolic charge distribution with 0.13 𝑚𝑚 rms length (0.6 𝑚𝑚

total length) is presented in Figure 4.18 (dashed line shows bunch shape). Bunch damping

energy modulation with linear ramp of the average energy deviation is observed.

- 87 -

Figure 4.18: Parabolic bunch wake potential.

Bunch shape change in various drift space lengths after 1 𝑚 ICMT is shown in

Figure19. The initial parabolic bunch transforms into several microbunches.

a

b

c

d

Figure 4.19: Line charge distribution after passing 𝑙𝑑𝑟𝑖𝑓𝑡 distance in a drift space

𝑎) 𝑙𝑑𝑟𝑖𝑓𝑡 = 3 𝑚, b) 𝑙𝑑𝑟𝑖𝑓𝑡 = 5 𝑚, c) 𝑙𝑑𝑟𝑖𝑓𝑡 = 7 𝑚, d) 𝑙𝑑𝑟𝑖𝑓𝑡 = 9 𝑚

- 88 -

The best bunching is on 7 𝑚 drift space length. On that distance FWHM of the first

microbunch is 7.5 𝜇𝑚. 1.8 % of total charge is concentrated in a full width region of the first

microbunch, and 2.2 % in three full width region. FWHM of the second microbunch is 3 𝜇𝑚.

1.54 % of total charge is concentrated in a full width region and 2.42 % in three full width

region.

4.6 Summary

Energy and charge density modulation of relativistic electron bunch after its

interaction with cold plasma and ICMT have been studied. The bunching and

microbunching processes of cold, low energy (10 𝑀𝑒𝑉) electron bunches of Gaussian,

rectangular and parabolic charge distributions are researched by ballistic method. It is

shown, that the Gaussian bunch interaction with these structures leads to its compression.

Bunch shape 𝑞0 𝑈0 𝜎0 𝜎𝑝, 𝑞𝑝 𝜎𝑡, 𝑞𝑡

Gaussian 100 𝑝𝐶 10 𝑀𝑒𝑉 100 𝜇𝑚 12 𝜇𝑚, 70 𝑝𝐶 8.02 𝜇𝑚, 70 𝑝𝐶

Table 1.1: Gaussian bunch parameters before and after interaction with plasma and ICMT.

Table 1.1 shows Gaussian bunch compression after plasma channel and ICMT. In

the table 𝑞0 is for bunch initial charge, 𝑈0 for initial energy, 𝜎0 is initial rms length, 𝜎𝑝 and

𝑞𝑝 are compressed bunch rms length and charge for plasma channel and finally 𝜎𝑡 and 𝑞𝑡

are rms length and charge for ICMT.

In a cold plasma a Gaussian bunch with 0.1 𝑚𝑚 rms length and 100 𝑝𝐶 charge can

be compressed by factor of 8, while in ICMT it can be compressed by factor of 12. For the

rectangular and parabolic bunches several times longer than the structure excited

frequency the microbunching process can be obtained.

Plasma ICMT

Bunch shape 𝑎0,𝑝 𝑎𝑝, 𝑞𝑝 𝑎0,𝑡 𝑎𝑡, 𝑞𝑡 𝑎𝑡, 𝑞𝑡

Rectangular 2 𝑚𝑚 40 𝜇𝑚, 14 𝑝𝐶 0.6 𝑚𝑚 8 𝜇𝑚, 7.8 𝑝𝐶 16 𝜇𝑚, 6 𝑝𝐶

Parabolic 2 𝑚𝑚 20 𝜇𝑚, 10 𝑝𝐶 0.6 𝑚𝑚 7.5 𝜇𝑚, 1.8 𝑝𝐶 3 𝜇𝑚, 1.5 𝑝𝐶

Table 1.2: Rectangular and parabolic bunches’ parameters before and after interaction with plasma

and ICMT.

- 89 -

Table 1.2 shows microbunching results for rectangular and parabolic bunches. In

the table indexes 𝑝 and 𝑡 are for plasma channel and ICMT, respectively. 𝑎0 is for bunch’s

initial length (FWHM), 𝑎and 𝑞 represents microbunches’ lengths and charges.

It is shown that in a cold plasma a uniform bunch with 2 𝑚𝑚 total length (0.58 𝑚𝑚

rms) and 100 𝑝𝐶 charge can be splitted into three microbunches with 40 𝜇𝑚 FWHM length

and 14 𝑝𝐶 charge. Parabolic bunch with 2 𝑚𝑚 full length ( 0.45 𝑚𝑚 rms) and 100 𝑝𝐶

charge forms three microbunches of 20 𝜇𝑚 full width and 10 𝑝𝐶 charges. Study of

microbunching process of rectangular bunch in ICMT shows that it is possible to generate

microbunches of 8 𝜇𝑚 FWHM length with 7.8 𝑝𝐶 charge and 16 𝜇𝑚 FWHM length with

5.94 𝑝𝐶 charge. Having an initial parabolic charge distribution, it is possible to generate

microbunches of 7.5 𝜇𝑚 FWHM length with 1.8 𝑝𝐶 charge and 3 𝜇𝑚 FWHM length with

1.54 𝑝𝐶 charge.

- 90 -

Summary

It is shown that in single-mode structures sub-ps bunches can be formed from

longer ones. The sub-ps bunch formation is researched in plasma channels and internally

coated metallic tubes. The ballistic bunching method is used in research. Bunch shape is

reconstructed numerically.

It is shown that two-layer structures with inner low conductivity material is a high

frequency single-mode structures. The theory of flat two-layer structure is developed. It is

shown that electrodynamic properties of the flat structure are similar to properties of the

cylindrical one. Rectangular cavity with horizontal two-layer metallic walls is studied and

correlation between modes obtained experimentally and theoretically is performed.

The main results of the dissertation are the following:

Matrix formalism has been developed which allows to couple point charge

radiated electromagnetic fields in the inner and outer regions of multilayer

parallel infinite plates.

Non-ultrarelativistic point charge excited electromagnetic fields in multilayer

parallel infinite plates are obtained analytically.

Explicit expression of non-ultrarelativistic point charge radiation fields in two-

layer parallel infinite plates with unbounded external walls is derived analytically.

Also point charge radiation fields in single-layer unbounded structure is derived

as a special case of two-layer structure.

Longitudinal impedance and dispersion relations of symmetrical two-layer

parallel infinite plates are derived. It is shown that, for low-conductivity thin inner

layer and high conductivity thick outer layer, the longitudinal impedance has a

narrow-band resonance in high frequency region.

- 91 -

The resonant frequency dependence of two-layer parallel infinite plates’

parameters is studied and resonance frequency dependence on structure

parameters is derived empirically.

Longitudinal wakefields generated by a point charge traveling through the center

of the two-layer parallel infinite plates with outer perfectly conducting material

are calculated. The longitudinal wake potential is a quasi-periodic function with

period given by the resonant frequency as c

𝑓𝑟𝑒𝑠.

Resonance frequencies of rectangular cavity with horizontal laminated walls are

obtained analytically.

It is shown that for rectangular cavity with vertical dimensions much bigger then

horizontal ones, the resonance frequencies are in good agreement with

frequencies of two-layer parallel infinite plates.

The measured resonances of copper cavity with horizontal walls internally

covered by germanium thin layer are correlated with analytically obtained

resonance modes of rectangular cavity. All resonances of the test cavity are in

conformity with analytically obtained modes.

It is shown that for appropriate structure parameters the rectangular cavity with

internally covered horizontal walls is a good candidate for bunch acceleration

and sub-ps (micro) bunch generation.

Compression of Gaussian bunches and microbunching of rectangular and

parabolic bunches is studied numerically in plasma channels and internally

coated metallic structures, based on the ballistic bunching method.

It is shown that Gaussian distributed bunches with initial 100 𝜇𝑚 rms length can

be compressed to 12 𝜇𝑚 and 8 𝜇𝑚 rms length in plasma channel and ICMT,

respectively.

- 92 -

It is shown that bunches with 20 𝜇𝑚 and 3 𝜇𝑚 full length can be formed due to

microbunching of parabolic bunches in plasma channel and ICMT, respectively.

It is shown that bunches with 40 𝜇𝑚 and 8 𝜇𝑚 full length can be formed due to

microbunching of rectangular bunches in plasma channel and ICMT,

respectively.

The results of the study can be useful for the development of new accelerating

structures for particle acceleration, monochromatic coherent radiation sources, ultrashort

bunch generation.

- 93 -

Acknowledgements

I express sincere gratitude to the director of CANDLE prof. Tsakanov, the head of the

doctor's work. M.I. Ivanyan and all the staff of CANDLE for their constant support and

assistance.

- 94 -

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